Multiplying Polynomials

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Name Class Date

Explore Analyzing a Visual Model for Polynomial Multiplication

The volume of a rectangular prism is the product of the length width and height of that prism If the dimensions are all known then the volume is a simple calculation What if some of the dimensions are given as binomials A binomial is a polynomial with two terms How would you find the volume of a rectangular prism that is x + 3 units long x + 2 units wide and xunits high The images below show two methods for finding the solution

V = length times width times height = (x + 3) (x + 2) x

x + 3

x

x + 2

A The first model shows the rectangular prism and its volume is calculated directly as the product of two binomials and a monomial

B The second image divides the rectangular prism into smaller prisms the dimensions of which are each

C The volume of a cube ( V 1 ) where all sides have a length of x is

D The volume of a rectangular prism ( V 2 ) with dimensions x by x by 2 is

E The volume of a rectangular prism ( V 3 ) with dimensions x by x by 3 is

F The volume of a rectangular prism ( V 4 ) with dimensions x by 3 by 2 is

G So the volume of the rectangular prism is the sum of the volumes of the four smaller regions

V 1 + V 2 + V 3 + V 4 = + + +

=

Resource Locker

xx

32

x

v = v1 + v2 + v3 + v4

v4 = volume of this piece




fourmonomials

x middot x middot x = x 3

x middot x middot 2 = 2x 2


x middot 3 middot 2 = 6x

x 3 2x 2 3x 2 6x

x 3 + 5 x 2 + 6x

Module 6 283 Lesson 2

62 Multiplying PolynomialsEssential question How do you multiply polynomials and what type of expression is the

result

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A2_MNLESE385894_U3M06L2 283 62714 937 PM

Common Core Math StandardsThe student is expected to

A-APRA1

Understand that polynomials form a system closed under multiplication hellip multiply polynomials Also A-APRC4 F-BFA1b

Mathematical Practices

MP4 Modeling

Language ObjectiveWork in pairs to complete a compare and contrast chart for addingsubtracting and multiplying polynomials

COMMONCORE

COMMONCORE

HARDCOVER PAGES 207214

Turn to these pages to find this lesson in the hardcover student edition

Multiplying Polynomials

ENGAGE Essential Question How do you multiply polynomials and what type of expression is the resultPossible answer You multiply two polynomials by

multiplying each term of one polynomial with each

term of the other polynomial The product is

another polynomial

PREVIEW LESSON PERFORMANCE TASKView the Engage section online Discuss the photo and how the total amount of oil produced is a function of both the number of wells and the amount produced by each well Then preview the Lesson Performance Task 283

HARDCOVER PAGES 207214

Turn to these pages to find this lesson in the hardcover student edition

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omp

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Name

Class Date

Explore Analyzing a Visual Model for Polynomial

Multiplication

The volume of a rectangular prism is the product of the length width and height of that prism

If the dimensions are all known then the volume is a simple calculation What if some of the

dimensions are given as binomials A binomial is a polynomial with two terms How would

you find the volume of a rectangular prism that is x + 3 units long x + 2 units wide and x

units high The images below show two methods for finding the solution

V = length times width times height

= (x + 3) (x + 2) x

x + 3

x

x + 2

The first model shows the rectangular prism and its volume is

calculated directly as the product of two binomials and a monomial

The second image divides the rectangular prism into

smaller prisms the dimensions of which are each

The volume of a cube ( V 1 ) where all sides have a length of x

is

The volume of a rectangular prism ( V 2 ) with dimensions x by x by 2

is

The volume of a rectangular prism ( V 3 ) with dimensions x by x by 3

is

The volume of a rectangular prism ( V 4 ) with dimensions x by 3 by 2

is

So the volume of the rectangular prism is the sum of the volumes of the

four smaller regions

V 1 + V 2 + V 3 + V 4 = + + +

=

Resource

Locker

A-APRA1 Understand that polynomials form a system hellip closed under hellip multiplication hellipmultiply

polynomials Also A-APRC4 F-BFA1bCOMMON

CORE

xx

3

2

x

v = v1 + v2 + v3 + v4

v4 = volume of

this piece

v3 = volume of

this piecev2 = volume of

this piece

v1 = volume of

this piece

four

monomials

x middot x middot x = x 3



x middot 3 middot 2 = 6x

x 3 2x 2 3x 2 6x

x 3 + 5 x 2 + 6x

Module 6

283

Lesson 2

6 2 Multiplying Polynomials

Essential Question How do you multiply polynomials and what type of expression is the

result

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CorrectionKey=NL-BCA-B

A2_MNLESE385894_U3M06L2 283

62714 936 PM

283 Lesson 6 2

L E S S O N 6 2

DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D

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Reflect

1 If all three dimensions were binomials how many regions would the rectangular prism be divided into

2 Discussion Can this method be applied to finding the volume of other simple solids Are there solids that this process would be difficult to apply to Are there any solids that this method cannot be applied to

Explain 1 Multiplying PolynomialsMultiplying polynomials involves using the product rule for exponents and the distributive property The product of two monomials is the product of the coefficients and the sum of the exponents of each variable

5x middot 6 x 3 = 30 x 1 + 3 -2 x 2 y 4 z middot 5 y 2 z = -10 x 2 y 4 + 2 z 1 + 1

= 30 x 4 = -10 x 2 y 6 z 2

When multiplying two binomials the distributive property is used Each term of one polynomial must be multiplied by each term of the other

(2 + 3x) (1 + x) = 2 (1 + x) + 3x (x + 1)

= 2 (1) + 2 (x) + 3x (x) + 3x (1)

= 2 + 2x + 3 x 1 + 1 + 3x

= 2 + 5x + 3 x 2

The polynomial 2 + 5x + 3 x 2 is called a trinomial because it has three terms

Example 1 Perform the following polynomial multiplications

(x + 2) (1 - 4x + 2 x 2 ) Find the product by multiplying horizontally

(x + 2) (2 x 2 - 4x + 1) Write the polynomials in standard form

x (2 x 2 ) + x (-4x) + x (1) + 2 (2 x 2 ) + 2 (-4x) + 2 (1) Distribute the x and the 2

2 x 3 - 4 x 2 + x + 4 x 2 -8x + 2 Simplify

2 x 3 - 7x + 2 Combine like terms

Therefore (x + 2) (2 x 2 - 4x + 1) = 2 x 3 - 7x + 2

8

Yes It would be difficult to find the volume of any shape such as a pyramid that did not

subdivide into smaller iterations of that shape and did not stack together well It cannot

be applied for example to a sphere


DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-CCA-C


A2_MNLESE385894_U3M06L2 284 6815 1040 PM

EXPLORE Analyzing a Visual Model for Polynomial Multiplication

INTEGRATE TECHNOLOGYStudents have the option of completing the polynomial multiplication activity either in

the book or online

QUESTIONING STRATEGIESHow does a polynomial model the volume of a real-world figure like a rectangular prism with

variable dimensions The volume of a figure like a

rectangular prism can be written as a cubic

polynomial because the polynomial represents

three dimensions For the rectangular prism the

cubic may be a product of the dimensions

EXPLAIN 1 Multiplying Polynomials

AVOID COMMON ERRORSStudents often are unsure whether to multiply polynomials horizontally or vertically Point out that if the polynomials have many terms multiplying them vertically may prevent errors because vertical multiplication is familiar and the locations of the product terms are similar to place value in a numerical product If the polynomials have few terms multiplying horizontally may be more convenient as long as the student remembers to use the distributive property to multiply each term of one polynomial by all other terms of the other polynomial

PROFESSIONAL DEVELOPMENT

Math BackgroundIn the previous lesson students discovered that polynomials are closed under addition and subtraction In this lesson students learn that polynomials are also closed under multiplication the product of two polynomials is a polynomial When we multiply two nonzero integers p by q we say that the product is an integer pq such that each digit of q is multiplied by each digit of p and then the partial products are added Multiplication of polynomials is much the same Given polynomials P (x) and Q (x) where both P (x) and Q (x) ne 0 we can write P (x) sdotQ (x) = R (x) R (x) will be a simplified polynomial with like terms combined

Multiplying Polynomials 284


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(3x - 4) (2 + x - 7 x 2 ) Find the product by multiplying vertically

-7 x 2 + + 2

3x - 4 Write each polynomial in standard form

___

- 4x - 8 Multipy -4 and (-7 x 2 + x + 2)

+ 3 x 2 + 6x Multipy and (-7 x 2 + x + 2)

_____

-21 x 3 + + 2x - 8 Combine like terms

Therefore (3x - 4) (2 + x - 7 x 2 ) =

Your Turn

3 (3 + 2x) (4 - 7x + 5 x 2 )

4 (x - 6) (3 - 8x - 4 x 2 )

Explain 2 Modeling with Polynomial MultiplicationMany real-world situations can be modeled with polynomial functions Sometimes a situation will arise in which a model is needed that combines two quantities modeled by polynomial functions In this case the desired model would be the product of the two known models

Example 2 Find the polynomial function modeling the desired relationship

Mr Silva manages a manufacturing plant From 1990 through 2005 the number of units produced (in thousands) can be modeled by N (x) = 002 x 2 + 02x + 3 where x is the number of years since 1990 The average cost per unit (in dollars) can be modeled by C (x) = -0002 x 2 - 01x + 2 where x is the number of years since 1990 Write a polynomial T (x) that can be used to model Mr Silvarsquos total manufacturing cost for those years

The total manufacturing cost is the product of the number of units made and the cost per unit

T (x) = N (x) middot C (x)

2x (5 x 2 ) + 2x (-7x) + 2x (4) + 3 (5 x 2 ) + 3 (-7x) + 3 (4) Distribute the 2x and the 3

10 x 3 - 14 x 2 + 8x + 15 x 2 - 21x + 12 Simplify

10 x 3 + x 2 - 13x + 12 Combine like terms

x (-4 x 2 ) + x (-8x) + x (3) - 6 (-4 x 2 ) - 6 (-8x) - 6 (3) Distribute the x and the -6

-4 x 3 - 8 x 2 + 3x + 24 x 2 + 48x - 18 Simplify

-4 x 3 + 16x 2 + 51x - 18 Combine like terms

-21 x 3 + 31 x 2 + 2x - 8

x

3x

28 x 2

-21 x 3

31 x 2



A2_MNLESE385894_U3M06L2 285 6815 1040 PM

1 2 3

COLLABORATIVE LEARNING

Small Group ActivityHave groups of students describe how to multiply polynomials Ask them to write an example multiplication problem in a graphic organizer similar to the one shown Students then pass the organizer to another student who writes in the next step and describes it They continue to pass the organizers until each problem is solved and all steps are explained A Sample organizer is shown

QUESTIONING STRATEGIESIs the commutative property of multiplication true for the multiplication of polynomials

Explain Yes the product will be the same

regardless of the order in which polynomials are

multiplied

After you have multiplied two polynomials how can you make sure you have not missed

any terms in the process Before simplifying the

product of a polynomial with m terms and a

polynomial with n terms has mn terms so count the

number of terms in the product

EXPLAIN 2 Modeling with Polynomial Multiplication

INTEGRATE MATHEMATICAL PRACTICESFocus on PatternsMP8 Point out that using a table with color to organize the products may be helpful when finding the product of real-world polynomials For example to find ( x 2 + 3x - 5) ( x 2 - x + 1) the table below might be used with the terms of each trinomial either above the columns or alongside the rows

x 2 -x +1

x 2 x 4 -x 3 + x 2

+3x +3 x 3 -3 x 2 +3x

-5 -5 x 2 +5x -5

Like terms shown with the same color are combined to complete the product

285 Lesson 6 2


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Multiply the two polynomials

002 x 2 + 02x + 3

times - 0002 x 2 - 01x + 2

――― 004 x 2 + 04x + 6

-0002 x 3 - 002 x 2 - 03x -000004 x 4 - 00004 x 3 - 0006 x 2

――――― -000004 x 4 - 00024 x 3 + 0014 x 2 + 01x + 6

Therefore the total manufacturing cost can be modeled by the following polynomial where x is the number of years since 1990

T (x) = -000004 x 4 - 00024 x 3 + 0014 x 2 + 01x + 6

B Ms Liao runs a small dress company From 1995 through 2005 the number of dresses she made can be modeled by N (x) = 03 x 2 - 16x + 14 and the average cost to make each dress can be modeled by C (x) = -0001 x 2 - 006x + 83 where x is the number of years since 1995 Write a polynomial that can be used to model Ms Liaorsquos total dressmaking costs T (x) for those years

The total dressmaking cost is the product of the number of dresses made and the cost per dress



03 x 2 - 16x + 14

times - 0001 x 2 + 83

―――― 249 x 2 - 1328x

-0018 x 3 - 084x

-00003 x + 00016 x 3 - 0014 x 2

――――――― -00003 x - 00164 x 3 + 2572 x 2 + 1162

Therefore the total dressmaking cost can be modeled by the following polynomial where x is the number of years since 1995

T (x) =

-006x

+1162

-00003 x 4 - 00164 x 3 + 2572 x 2 - 1412x + 1162

4

4

+0096 x 2

-1412x


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A2_MNLESE385894_U3M06L2indd 286 31914 1221 PM

QUESTIONING STRATEGIESWhat property of exponents is used to find the partial products When you multiply two

powers with the same base you add their

exponents

DIFFERENTIATE INSTRUCTION

Multiple RepresentationsHave students work in small groups to multiply two polynomials such as ( x 2 + 3x - 5) ( x 2 - x + 1) Each student in the group should choose a different method such as multiplying horizontally multiplying vertically or using a table Have students discuss the ways in which the methods are alike and the ways in which they differ



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Your Turn

5 Brent runs a small toy store specializing in wooden toys From 2000 through 2012 the number of toys Brent made can be modeled by N (x) = 07 x 2 - 2x + 23 and the average cost to make each toy can be modeled by C (x) = -0004 x 2 - 008x + 25 where x is the number of years since 2000 Write a polynomial that can be used to model Brentrsquos total cost for making the toys T (x) for those years

Explain 3 Verifying Polynomial IdentitiesYou have already seen certain special polynomial relationships For example a difference of two squares can be easily factored x 2 - a 2 = (x + a) (x - a) This equation is an example of a polynomial identity a mathematical relationship equating one polynomial quantity to another Another example of a polynomial identity is

(x + a) 2 - (x - a) 2 = 4ax

The identity can be verified by simplifying one side of the equation to match the other

Example 3 Verify the given polynomial identity

(x + a) 2 - (x - a) 2 = 4ax

The right side of the identity is already fully simplified Simplify the left-hand side

(x + a) 2 - (x - a) 2 = 4ax

x 2 + 2ax + a 2 - ( x 2 - 2ax + a 2 ) = 4ax Square each binomial x 2 + 2ax + a 2 - x 2 + 2ax - a 2 = 4ax Distribute the negative

x 2 - x 2 + 2ax + 2ax + a 2 - a 2 = 4ax Rearrange terms

4ax = 4ax Simplify

Therefore (x + a) 2 - (x - a) 2 = 4ax is a true statement

The total cost is the product of the number of toys made and the cost per toy


07 x 2 - 2x + 23

times - 0004 x 2 - 008x + 25

――――― 175 x 2 - 50x + 575

-0056 x 3 + 016 x 2 - 184x

-00028 x 4 +thinsp 0008 x 3 - 0092 x 2

―――――― -00028 x 4 - 0048 x 3 + 17568 x 2 - 5184x + 575

Therefore the total cost of making the toys can be modeled by the following polynomial

where x is the number of years since 2000

T (x) = -00028 x 4 - 0048 x 3 + 17568 x 2 - 5184x + 575



A2_MNLESE385894_U3M06L2 287 61615 839 AM

EXPLAIN 3 Verifying Polynomial Identities

AVOID COMMON ERRORSStudents may think that they need to analyze each side of a polynomial equation in order to verify that the equation expresses a polynomial identity Point out that if one side of the equation is a monomial then that side is complete There may be more than one way to proceed but the arithmetic operations must be performed on both sides if necessary until the two sides match

QUESTIONING STRATEGIESHow do you verify a polynomial identity You

perform the operations indicated on each

side of the identity until the two sides match

LANGUAGE SUPPORT

Communicate MathHave students complete a chart like the following showing similarities and differences

Operation Add and Subtract Polynomials

Multiply Polynomials

Alike The result is another polynomial

The result is another polynomial

Different You can only add and subtract like terms

You donrsquot need to multiply like terms

287 Lesson 6 2


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(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3

a ( a 2 ) + a ( ) + a ( b 2 ) + b ( a 2 ) + (-ab) + b ( b 2 ) = a 3 + b 3 Distribute a and b

a 3 - a 2 b + a b 2 + - a b 2 + = a 3 + b 3

a 3 - + a 2 b + a b 2 - + b 3 = a 3 + b 3 Rearrange terms

a 3 b 3 = a 3 + b 3 Combine like terms

Therefore (a + b) ( a 2 - ab + b 2 ) = a 3 + b 3 is a statement

Your Turn

6 Show that a 5 - b 5 = (a - b) ( a 4 + a 3 b + a 2 b 2 + a b 3 + b 4 )

7 Show that (a - b) ( a 2 + ab + b 2 ) = a 3 - b 3

Explain 4 Using Polynomial IdentitiesThe most obvious use for polynomial identities is simplifying algebraic expressions but polynomial identities often turn out to have nonintuitive uses as well

Example 4 For each situation find the solution using the given polynomial identity

The polynomial identity ( x 2 + y 2 ) 2 = ( x 2 - y 2 ) 2 + (2xy) 2 can be used to identify Pythagorean triples Generate a Pythagorean triple using x = 4 and y = 3

Substitute the given values into the identity

( 4 2 + 3 2 ) 2 = ( 4 2 - 3 2 ) 2 + (2 middot 4 middot 3) 2 (16 + 9) 2 = (16 - 9) 2 + (24) 2 (25) 2 = (7) 2 + (24) 2 625 = 49 + 576

625 = 625

Therefore 7 24 25 is a Pythagorean triple

The left side of the identity is already fully simplified Simplify the right-hand side a 5 - b 5 = a ( a 4 ) + a ( a 3 b) + a ( a 2 b 2 ) + a ( ab 3 ) + a ( b 4 ) - b ( a 4 ) - b ( a 3 b) - b ( a 2 b 2 ) - b ( ab 3 )

- b ( b 4 ) a 5 - b 5 = a 5 + a 4 b + a 3 b 2 + a 2 b 3 + a b 4 - a 4 b - a 3 b 2 - a 2 b 3 - a b 4 - b 5 a 5 - b 5 = a 5 - b 5

The right side of the identity is already fully simplified Simplify the left-hand side a ( a 2 ) + a (ab) + a ( b 2 ) - b ( a 2 ) - b (ab) - b ( b 2 ) = a 3 - b 3 Distribute a and b a 3 + a 2 b + a b 2 - a 2 b - a b 2 - b 3 = a 3 - b 3 Simplify a 3 - b 3 = a 3 - b 3 Combine like terms

-ab

a 2 b

a 2 b Simplify

true

a b 2

b 3

+

b


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A2_MNLESE385894_U3M06L2 288 62714 118 PM

EXPLAIN 4 Using Polynomial Identities

CONNECT VOCABULARY Students may not understand identity in the context of using polynomialsrsquo identities Tell them that once an identity is established they should then apply the identity to numbers much as they would apply a known formula to a geometric figure In the process of using the identity they do not re-verify the identity

QUESTIONING STRATEGIESHow are polynomial identities used They may be used to simplify algebraic

expressions or to find shortcuts for

polynomial-based formulas or mental math

calculations



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B The identity (x + y) 2 = x 2 + 2xy + y 2 can be used for mental-math calculations to quickly square numbers

Find the square of 27

Find two numbers whose sum is equal to 27

Let x = and y = 7

Evaluate

(20 + ) 2

= 2 0 2 + + 7 2

2 7 2 = 400 + + 49

2 7 2 =

Verify by using a calculator to find 2 7 2

2 7 2 =

Your Turn

8 The identity (x + y) (x - y) = x 2 - y 2 can be used for mental-math calculations to quickly multiply two numbers in specific situations

Find the product of 37 and 43 (Hint What values should you choose for x and y so the equation calculates the product of 37 and 43)

9 The identity (x - y) 2 = x 2 - 2xy + y 2 can also be used for mental-math calculations to quickly square numbers

Find the square of 18 (Hint What values should you choose for x and y so the equation calculates the square of 18)

20

7 2 middot 20 middot 7

280

729

729

Substitute x = 40 and y = 3 into the identity and evaluate

(40 + 3) (40 - 3) = 40 2 - 3 2

43 37 = 1600 - 9

43 37 = 1591

Possible answer 20 - 2 = 18


(20 - 2) 2 = 20 2 - 2 2 20 + 2 2

= 400 - 80 + 4

= 324



A2_MNLESE385894_U3M06L2 289 161014 838 AM

AVOID COMMON ERRORSRegardless of the method students use to multiply polynomials a common error is to use the properties of exponents incorrectly multiplying exponents that should be added Remind students that the product of two powers with the same base is the base raised to the sum of the powers or b m bull b n = b m + n

289 Lesson 6 2


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Elaborate

10 What property is employed in the process of polynomial multiplication

11 How can you use unit analysis to justify multiplying two polynomial models of real-world quantities

12 Give an example of a polynomial identity and how itrsquos useful

13 Essential Question Check-In When multiplying polynomials what type of expression is the product

Evaluate Homework and Practice

1 The dimensions for a rectangular prism are x + 5 for the length x + 1 for the width and x for the height What is the volume of the prism

Perform the following polynomial multiplications

2 (3x - 2) (2 x 2 + 3x - 1)

3 ( x 3 + 3 x 2 + 1) (3 x 2 + 6x - 2)

bull Online Homeworkbull Hints and Helpbull Extra Practice

The distributive property

x 3 +3 x 2 +1

times 3 x 2 +6x -2

――――― -2 x 3 -6 x 2 -2 Multiply -2 and ( x 3 + 3 x 2 + 1)

6 x 4 +18 x 3 +6x Multiply 6x and ( x 3 + 3 x 2 + 1)

3 x 5 + 9 x 4 +3 x 2 Multiply 3 x 2 and ( x 3 + 3 x 2 + 1)

――――― 3 x 5 + 15 x 4 +16 x 3 -3 x 2 +6x -2 Combine like terms

The units of the polynomials need to combine in such a way that their product is the

desired unit

See student work answers will vary

A polynomial

(x + 5) (x + 1) x = x 3 + 6 x 2 + 5x

3x (2 x 2 ) + 3x (3x) + 3x (-1) - 2 (2 x 2 ) - 2 (3x) - 2 (-1) Distribute the 3x and the -2

6 x 3 + 9 x 2 - 3x - 4 x 2 - 6x +2 Simplify

6 x 3 + 5 x 2 - 9x + 2 Combine like terms


DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-ACA-A


A2_MNLESE385894_U3M06L2indd 290 31914 1221 PMExercise Depth of Knowledge (DOK)COMMONCORE Mathematical Practices

1 1 Recall of Information MP6 Precision

2ndash7 1 Recall of Information MP6 Precision

8ndash11 2 SkillsConcepts MP4 Modeling

12ndash21 2 SkillsConcepts MP2 Reasoning

22 3 Strategic Thinking MP2 Reasoning

23 3 Strategic Thinking MP5 Using Tools

ELABORATE QUESTIONING STRATEGIES

How is the distributive property used to multiply two polynomials Each monomial

term of one polynomial must be multiplied by each

term of the other polynomial so the distributive

property applies

CONNECT VOCABULARY Relate the prefixes bi- and tri- to the meaning of binomial (two terms) and trinomial (three terms)

SUMMARIZE THE LESSONWhat points should you remember when multiplying polynomials Use the

distributive property to multiply every term of one

polynomial by every term of the other polynomial

combine like terms and align like terms

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreAnalyzing a Visual Model for Polynomial Multiplication

Exercise 1

Example 1Multiplying Polynomials

Exercises 2ndash7

Example 2Modeling with Polynomial Multiplication

Exercises 8ndash11

Example 3Verifying Polynomial Identities

Exercises 12ndash15

Example 4Using Polynomial Identities

Exercises 16ndash21



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4 ( x 2 + 9x + 7) (3 x 2 + 9x + 5)

5 (2x + 5y) (3 x 2 - 4xy + 2 y 2 )

6 ( x 3 + x 2 + 1) ( x 2 - x - 5)

7 (4 x 2 + 3x + 2) (3 x 2 + 2x - 1)

x 2 +9x +7

times 3 x 2 +9x +5

――――― 5 x 2 +45x +35 Multiply 5 and ( x 2 + 9x + 7)

9 x 3 +81 x 2 +63x Multiply 9x and ( x 2 + 9x + 7)

3x 4 + 27 x 3 +21 x 2 Multiply 3 x 2 and ( x 2 + 9x + 7)

――――― 3 x 4 + 36 x 3 +107 x 2 +108x +35 Combine like terms

2x (3 x 2 ) + 2x (-4xy) + 2x (2 y 2 ) + 5y (3 x 2 ) + 5y (-4xy) + 5y (2 y 2 ) Distribute the 2x and the 5y

6 x 3 - 8 x 2 y + 4x y 2 + 15 x 2 y - 20x y 2 + 10 y 3 Simplify

6 x 3 + 7 x 2 y - 16x y 2 + 10 y 3 Combine like terms

x 3 + x 2 +1

times x 2 -x -5

―――― -5 x 3 -5 x 2 -5 Multiply -5 and ( x 3 + x 2 + 1)

- x 4 - x 3 -x Multiply -x and ( x 3 + x 2 + 1)

x 5 + x 4 x 2 Multiply x 2 and ( x 3 + x 2 + 1)

―――― x 5 -6 x 3 -4 x 2 -x -5 Combine like terms

4 x 2 + 3x +2

times 3 x 2 + 2x -1

―――― -4 x 2 - 3x -2 Multiply -1 and (4 x 2 + 3x + 2)

8 x 3 +6 x 2 + 4x Multiply 2x and (4 x 2 + 3x + 2)

12 x 4 +9 x 3 +6 x 2 Multiply 3 x 2 and (4 x 2 + 3x + 2)

―――― 12 x 4 +17 x 3 +8 x 2 +x -2 Combine like terms



A2_MNLESE385894_U3M06L2 291 62814 132 PMExercise Depth of Knowledge (DOK)COMMONCORE Mathematical Practices


25 3 Strategic Thinking MP3 Logic


INTEGRATE TECHNOLOGYPoint out that when multiplying two polynomials in one variable students can

graph the expressions before they are multiplied and again after they are multiplied The graphs should be coincident

291 Lesson 6 2


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Write a polynomial function to represent the new value

8 The volume of a stock or number of shares traded is modeled over time during a given day by S (x) = x 5 - 3 x 4 + 10 x 2 - 6x + 30 The cost per share of that stock during that day is modeled by C (x) = 0004 x 4 - 002 x 2 + 03x + 4 Write a polynomial function V(x) to model the changing value during that day of the trades made of shares of that stock

9 A businessman models the number of items (in thousands) that his company sold from 1998 through 2004 as N (x) = -01 x 3 + x 2 - 3x + 4 and the average price per item (in dollars) as P (x) = 02x + 5 where x represents the number of years since 1998 Write a polynomial R (x) that can be used to model the total revenue for this company

10 Biology A biologist has found that the number of branches on a certain rare tree can be modeled by the polynomial b (y) = 4 y 2 + y where y is the number of years after the tree reaches a height of 6 feet The number of leaves on each branch can be modeled by the polynomial l (y) = 2 y 3 + 3 y 2 + y Write a polynomial describing the total number of leaves on the tree

The total revenue will be the product of the number of items sold and the price at which each item is sold


R (x) = (02x + 5) (-01 x 3 + x 2 - 3x + 4)

= 02x (-01 x 3 ) + 02x ( x 2 ) + 02x (-3x) + 02x (4) + 5 (-01 x 3 ) + 5 ( x 2 ) + 5 (-3x) + 5 (4)

= -002 x 4 + 02 x 3 - 0 6x 2 + 08x - 05x 3 + 5x 2 - 15x + 20

=-002 x 4 - 0 3x 3 + 44 x 2 - 142x + 20

The value is equal to the number of shares traded times the cost per share

V (x) = ( x 5 - 3 x 4 + 10 x 2 - 6x + 30) (0004 x 4 - 002 x 2 + 03x + 4)

x 5 -3 x 4 +10 x 2 -6x +30

times 0004 x 4 -002 x 2 +03x +thinsp4 4 x 5 -12 x 4 +40 x 2 -24x +120

03 x 6 -09 x 5 +3 x 3 -18 x 2 +9x

-002 x 7 +006 x 6 -02 x 4 +012 x 3 -06 x 2

0004 x 9 -0012 x 8 +004 x 6 -0024 x 5 +012 x 4

――――――――――― 0004 x 9 -0012 x 8 -002 x 7 +04 x 6 +3076 x 5 -1208 x 4 +312 x 3 +376 x 2 -15x + 120

V (x) = 0004 x 4 - 0012 x 8 - 002 x 7 + 04 x 6 + 3076 x 5 - 1208 x 4 + 312 x 3 + 376 x 2 -15x + 120

T (y) = (4 y 2 + y) (2 y 3 + 3 y 2 + y)

= 8 y 5 + 12 y 4 + 4 y 3 + 2 y 4 + 3 y 3 + y 2

= 8 y 5 + 14 y 4 + 7 y 3 + y 2


DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-CCA-C


A2_MNLESE385894_U3M06L2 292 61615 840 AM

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP3 Help students clarify how to multiply polynomials by having them work in groups Have one student complete one step of the multiplication process including an explanation of the process then pass the problem to another student who completes the second step including an explanation Continue passing the problem until it is complete



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11 Physics An object thrown in the air has a velocity after t seconds that can be described by v (t) = -98t + 24 (in meterssecond) and a height h (t) = -49 t 2 + 24t + 60 (in meters) The object has mass m = 2 kilograms The kinetic energy of the object is given by K = 1 __ 2 m v 2 and the potential energy is given by U = 98mh Find an expression for the total kinetic and potential energy K + U as a function of time What does this expression tell you about the energy of the falling object

Verify the given polynomial identity

12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)

= 1752Since the sum is a constant this means that the energy of the object is constant and that as it gains kinetic energy by falling it loses the same amount of potential energy


(x + y + z) 2 = x (x) + x (y) + x (z) + y (x) + y (y) + y (z) + z (x) + z (y) + z (z)

= x 2 + xy + xz + yx + y 2 + yz + zx + zy + z 2

= x 2 + y 2 + z 2 + xy + yx + xz + zx + yz + zy

= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

The left side of the identity is already fully simplified Simplify the right-hand side

a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5

The left side of the identity is already fully simplified Simplify the right-hand sideExamine (x - y) (x + y) ( x 2 + y 2 ) Recall that (x + y) (x - y) = x 2 - y 2 Substitute on the right side of the equation

x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4




MULTIPLE REPRESENTATIONSTo help students structure how to multiply polynomials have them use tables similar to the one shown below for ( x 2 + 2x + 1) ( x 3 + 3x - 2) These diagrams provide more visual support than the more standard vertical method Have students share their tables with a partner describing the patterns they see and telling how they got their product polynomials

x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2

Evaluate the following polynomials using one or more of these identities

(x + y) 2 = x 2 + 2xy + y 2 (x + y) (x - y) = x 2 - y 2 or (x - y) 2 = x 2 - 2xy + y 2

16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68

22 Explain the Error A polynomial identity for the difference of two cubes is a3 - b3 = (a - b)(a2 + ab + b2) A student uses the identity to factor 64 - 27x6 Identify the error the student made and then correct it

Each term of 64 - 27x6 is a perfect cube Let a = 4 and b = 3x2 Then 64 ndash 27x6 = 43 - (3x2)3 = (4 - 3x2)(42 + 4(-3x2) + (-3x2)2) = (4 - 3x2)(16 - 12x2 + 9x4)

a 2 ( x 2 ) + a 2 ( y 2 ) + b 2 ( x 2 ) + b 2 ( y 2 ) = (ax) 2 - 2 (ax) (by) + (by) 2 + (bx)

2 + 2 (bx) (ay) + (ay) 2

a 2 x 2 + a 2 y 2 + b 2 x 2 + b 2 y 2 = a 2 x 2 - 2axby + b 2 y 2 + b 2 x 2 + 2axby + a 2 y 2

a 2 x 2 + a 2 y 2 + b 2 x 2 + b 2 y 2 = a 2 x 2 + a 2 y 2 + b 2 x 2 + b 2 y 2

4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052

The student substituted -3 x 2 for b instead of 3 x 2 for b in the ab and b2

terms of the trinomial

64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209


DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D


A2_MNLESE385894_U3M06L2indd 294 3116 113 AM

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP4 When students multiply polynomials they may leave out some of the partial products Tell students to write down all of the partial products and circle the monomial in each one Then ask them to draw arrows from the monomial in the first polynomial to the matching monomial in the second polynomial

AVOID COMMON ERRORSWhen using the rules for some special products of polynomials students often forget to apply the power of a power property of exponents to the coefficients of terms in the polynomial Suggest that students first write the coefficient and variable within parentheses with the exponent applied to both and then simplify



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23 Determine how many terms there will be after performing the polynomial multiplication

a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4

HOT Focus on Higher Order Thinking

24 Multi-Step Given the polynomial identity x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 ) a Verify directly by expanding the right hand side

b Use another polynomial identity to verify this identity (Note that a 6 = ( a 2 ) 3 = ( a 3 ) 2 )

25 Communicate Mathematical Ideas Explain why the set of polynomials is closed under multiplication

26 Critical Thinking Explain why every other term of the polynomial product (x - y) 5 written in standard form is subtracted when (x - y) is raised to the fifth power

a (5x) (3x) = 15 x 2 1 term b (3x) (2x + 1) = 6 x 2 + 3x 2 terms c (x + 1) (x - 1) = x 2 - 1 2 terms d (x + 2) (3 x 2 - 2x + 1) = 3 x 3 + 4 x 2 - 3x + 2 4 terms

x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )

x 6 + y 6 = x 2 ( x 4 ) + x 2 (- x 2 y 2 ) + x 2 ( y 4 ) + y 2 ( x 4 ) + y 2 (- x 2 y 2 ) + y 2 ( y 4 )

x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6

Use a 6 = ( a 2 ) 3 to replace x 6 + y 6 with ( x 2 )

3 + ( y 2 )

3

Now use the identity for the sum of two cubes (a + b) ( a 2 - ab + b 2 ) = a 3 + b 3 to

simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )

When the power of -y in a term of the product is odd the term is subtracted and when the power of -y in a term of the product is even the term is added

Since a x m sdot b x n = ab x m + n for real numbers a and b and whole numbers m and n the product of two monomials is another monomial Therefore the product of two polynomials which are sums of monomials is again a sum of monomials which is another polynomial

X

X

X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM

PEER-TO-PEER DISCUSSIONInstruct one student in each pair to write two polynomials in standard form while the other student gives verbal instructions for multiplying the polynomials Then have students switch roles repeat the exercise and give instructions for multiplying two new polynomials

JOURNALHave students make a table describing the methods for multiplying polynomials Give examples for multiplying monomials binomials and trinomials as well as for verifying polynomial identities

295 Lesson 6 2


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Lesson Performance Task

The table presents data about oil wells in the state of Oklahoma from 1992 through 2008

Year Number of Wells Average Daily Oil Production per Well (Barrels)

2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299

a Given the data in this table use regression to find models for the number of producing wells (cubic regression) and average daily well output (quadratic regression) in terms of t years since 1992

b Find a function modeling the total daily oil output for the state of Oklahoma

a Producing oil wells W (t) = 0884 t 3 + 396 t 2 - 1510t + 93700

Average daily output (in barrels per well) O (t) = 000418 t 2 - 0125t + 301

where t equals time (1992 = 1 1993 = 2 etc)

b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037

To the correct number of significant figures

D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY

Have students research the price of oil per barrel for each year from 1992 to 2008 and use polynomial regression to find a model P (t) for the price of oil Then have students use the model and the daily output model O (t) they determined in the Performance Task to find a function modeling the total value of oil produced each day by the state of Oklahoma Ask students to calculate the daily oil income in 1992 and compare it to 2008

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP2 Ask students to look at the data table and describe any trends they see in number of wells and daily output over time Both decrease with

time Based on these trends have students predict the behavior of the total daily oil output over time It

will decrease Have students graph the total daily output function D (t) for the time period from 1992 to 2008 to test their predictions

AVOID COMMON ERRORSSome students may multiply exponents instead of adding them Polynomials have two types of numerical values exponents and coefficients In the term 0884 t 3 the coefficient is 0884 and the exponent is 3 Have students explain what to do with each value when multiplying two polynomials Multiply the coefficients and add the exponents

Scoring Rubric2 points Student correctly solves the problem and explains hisher reasoning1 point Student shows good understanding of the problem but does not fully solve or explain hisher reasoning0 points Student does not demonstrate understanding of the problem



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Reflect

1 If all three dimensions were binomials how many regions would the rectangular prism be divided into

2 Discussion Can this method be applied to finding the volume of other simple solids Are there solids that this process would be difficult to apply to Are there any solids that this method cannot be applied to

Explain 1 Multiplying PolynomialsMultiplying polynomials involves using the product rule for exponents and the distributive property The product of two monomials is the product of the coefficients and the sum of the exponents of each variable

5x middot 6 x 3 = 30 x 1 + 3 -2 x 2 y 4 z middot 5 y 2 z = -10 x 2 y 4 + 2 z 1 + 1

= 30 x 4 = -10 x 2 y 6 z 2

When multiplying two binomials the distributive property is used Each term of one polynomial must be multiplied by each term of the other

(2 + 3x) (1 + x) = 2 (1 + x) + 3x (x + 1)

= 2 (1) + 2 (x) + 3x (x) + 3x (1)

= 2 + 2x + 3 x 1 + 1 + 3x

= 2 + 5x + 3 x 2

The polynomial 2 + 5x + 3 x 2 is called a trinomial because it has three terms

Example 1 Perform the following polynomial multiplications

(x + 2) (1 - 4x + 2 x 2 ) Find the product by multiplying horizontally

(x + 2) (2 x 2 - 4x + 1) Write the polynomials in standard form

x (2 x 2 ) + x (-4x) + x (1) + 2 (2 x 2 ) + 2 (-4x) + 2 (1) Distribute the x and the 2

2 x 3 - 4 x 2 + x + 4 x 2 -8x + 2 Simplify

2 x 3 - 7x + 2 Combine like terms

Therefore (x + 2) (2 x 2 - 4x + 1) = 2 x 3 - 7x + 2

8

Yes It would be difficult to find the volume of any shape such as a pyramid that did not

subdivide into smaller iterations of that shape and did not stack together well It cannot

be applied for example to a sphere




A2_MNLESE385894_U3M06L2 284 6815 1040 PM

EXPLORE Analyzing a Visual Model for Polynomial Multiplication

INTEGRATE TECHNOLOGYStudents have the option of completing the polynomial multiplication activity either in

the book or online

QUESTIONING STRATEGIESHow does a polynomial model the volume of a real-world figure like a rectangular prism with

variable dimensions The volume of a figure like a

rectangular prism can be written as a cubic

polynomial because the polynomial represents

three dimensions For the rectangular prism the

cubic may be a product of the dimensions

EXPLAIN 1 Multiplying Polynomials

AVOID COMMON ERRORSStudents often are unsure whether to multiply polynomials horizontally or vertically Point out that if the polynomials have many terms multiplying them vertically may prevent errors because vertical multiplication is familiar and the locations of the product terms are similar to place value in a numerical product If the polynomials have few terms multiplying horizontally may be more convenient as long as the student remembers to use the distributive property to multiply each term of one polynomial by all other terms of the other polynomial

PROFESSIONAL DEVELOPMENT

Math BackgroundIn the previous lesson students discovered that polynomials are closed under addition and subtraction In this lesson students learn that polynomials are also closed under multiplication the product of two polynomials is a polynomial When we multiply two nonzero integers p by q we say that the product is an integer pq such that each digit of q is multiplied by each digit of p and then the partial products are added Multiplication of polynomials is much the same Given polynomials P (x) and Q (x) where both P (x) and Q (x) ne 0 we can write P (x) sdotQ (x) = R (x) R (x) will be a simplified polynomial with like terms combined



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-7 x 2 + + 2


___

- 4x - 8 Multipy -4 and (-7 x 2 + x + 2)

+ 3 x 2 + 6x Multipy and (-7 x 2 + x + 2)

_____


Therefore (3x - 4) (2 + x - 7 x 2 ) =

Your Turn

3 (3 + 2x) (4 - 7x + 5 x 2 )

4 (x - 6) (3 - 8x - 4 x 2 )







10 x 3 - 14 x 2 + 8x + 15 x 2 - 21x + 12 Simplify



-4 x 3 - 8 x 2 + 3x + 24 x 2 + 48x - 18 Simplify


-21 x 3 + 31 x 2 + 2x - 8

x

3x

28 x 2

-21 x 3

31 x 2



A2_MNLESE385894_U3M06L2 285 6815 1040 PM

1 2 3






multiplied








x 2 -x +1

x 2 x 4 -x 3 + x 2

+3x +3 x 3 -3 x 2 +3x

-5 -5 x 2 +5x -5


285 Lesson 6 2


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002 x 2 + 02x + 3

times - 0002 x 2 - 01x + 2

――― 004 x 2 + 04x + 6

-0002 x 3 - 002 x 2 - 03x -000004 x 4 - 00004 x 3 - 0006 x 2

――――― -000004 x 4 - 00024 x 3 + 0014 x 2 + 01x + 6


T (x) = -000004 x 4 - 00024 x 3 + 0014 x 2 + 01x + 6





03 x 2 - 16x + 14

times - 0001 x 2 + 83

―――― 249 x 2 - 1328x

-0018 x 3 - 084x

-00003 x + 00016 x 3 - 0014 x 2

――――――― -00003 x - 00164 x 3 + 2572 x 2 + 1162


T (x) =

-006x

+1162

-00003 x 4 - 00164 x 3 + 2572 x 2 - 1412x + 1162

4

4

+0096 x 2

-1412x







exponents





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Your Turn



(x + a) 2 - (x - a) 2 = 4ax



(x + a) 2 - (x - a) 2 = 4ax


(x + a) 2 - (x - a) 2 = 4ax



4ax = 4ax Simplify




07 x 2 - 2x + 23

times - 0004 x 2 - 008x + 25

――――― 175 x 2 - 50x + 575

-0056 x 3 + 016 x 2 - 184x

-00028 x 4 +thinsp 0008 x 3 - 0092 x 2

―――――― -00028 x 4 - 0048 x 3 + 17568 x 2 - 5184x + 575



T (x) = -00028 x 4 - 0048 x 3 + 17568 x 2 - 5184x + 575



A2_MNLESE385894_U3M06L2 287 61615 839 AM






LANGUAGE SUPPORT








287 Lesson 6 2


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(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


a 3 - a 2 b + a b 2 + - a b 2 + = a 3 + b 3




Your Turn







( 4 2 + 3 2 ) 2 = ( 4 2 - 3 2 ) 2 + (2 middot 4 middot 3) 2 (16 + 9) 2 = (16 - 9) 2 + (24) 2 (25) 2 = (7) 2 + (24) 2 625 = 49 + 576

625 = 625





-ab

a 2 b

a 2 b Simplify

true

a b 2

b 3

+

b




A2_MNLESE385894_U3M06L2 288 62714 118 PM






calculations



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Let x = and y = 7

Evaluate

(20 + ) 2

= 2 0 2 + + 7 2

2 7 2 = 400 + + 49

2 7 2 =


2 7 2 =

Your Turn





20


280

729

729


(40 + 3) (40 - 3) = 40 2 - 3 2

43 37 = 1600 - 9

43 37 = 1591



(20 - 2) 2 = 20 2 - 2 2 20 + 2 2

= 400 - 80 + 4

= 324



A2_MNLESE385894_U3M06L2 289 161014 838 AM


289 Lesson 6 2


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Elaborate








2 (3x - 2) (2 x 2 + 3x - 1)

3 ( x 3 + 3 x 2 + 1) (3 x 2 + 6x - 2)



x 3 +3 x 2 +1

times 3 x 2 +6x -2






desired unit


A polynomial

(x + 5) (x + 1) x = x 3 + 6 x 2 + 5x


6 x 3 + 9 x 2 - 3x - 4 x 2 - 6x +2 Simplify
















property applies






EVALUATE

ASSIGNMENT GUIDE



Exercise 1


Exercises 2ndash7


Exercises 8ndash11


Exercises 12ndash15


Exercises 16ndash21



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4 ( x 2 + 9x + 7) (3 x 2 + 9x + 5)

5 (2x + 5y) (3 x 2 - 4xy + 2 y 2 )

6 ( x 3 + x 2 + 1) ( x 2 - x - 5)

7 (4 x 2 + 3x + 2) (3 x 2 + 2x - 1)

x 2 +9x +7

times 3 x 2 +9x +5








x 3 + x 2 +1

times x 2 -x -5





4 x 2 + 3x +2

times 3 x 2 + 2x -1













291 Lesson 6 2


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R (x) = (02x + 5) (-01 x 3 + x 2 - 3x + 4)

= 02x (-01 x 3 ) + 02x ( x 2 ) + 02x (-3x) + 02x (4) + 5 (-01 x 3 ) + 5 ( x 2 ) + 5 (-3x) + 5 (4)

= -002 x 4 + 02 x 3 - 0 6x 2 + 08x - 05x 3 + 5x 2 - 15x + 20

=-002 x 4 - 0 3x 3 + 44 x 2 - 142x + 20


V (x) = ( x 5 - 3 x 4 + 10 x 2 - 6x + 30) (0004 x 4 - 002 x 2 + 03x + 4)

x 5 -3 x 4 +10 x 2 -6x +30


03 x 6 -09 x 5 +3 x 3 -18 x 2 +9x

-002 x 7 +006 x 6 -02 x 4 +012 x 3 -06 x 2

0004 x 9 -0012 x 8 +004 x 6 -0024 x 5 +012 x 4

――――――――――― 0004 x 9 -0012 x 8 -002 x 7 +04 x 6 +3076 x 5 -1208 x 4 +312 x 3 +376 x 2 -15x + 120

V (x) = 0004 x 4 - 0012 x 8 - 002 x 7 + 04 x 6 + 3076 x 5 - 1208 x 4 + 312 x 3 + 376 x 2 -15x + 120

T (y) = (4 y 2 + y) (2 y 3 + 3 y 2 + y)

= 8 y 5 + 12 y 4 + 4 y 3 + 2 y 4 + 3 y 3 + y 2

= 8 y 5 + 14 y 4 + 7 y 3 + y 2




A2_MNLESE385894_U3M06L2 292 61615 840 AM




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12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)






= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz


a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5


x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4





x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









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a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

X

X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


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2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









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-7 x 2 + + 2


___

- 4x - 8 Multipy -4 and (-7 x 2 + x + 2)

+ 3 x 2 + 6x Multipy and (-7 x 2 + x + 2)

_____


Therefore (3x - 4) (2 + x - 7 x 2 ) =

Your Turn

3 (3 + 2x) (4 - 7x + 5 x 2 )

4 (x - 6) (3 - 8x - 4 x 2 )







10 x 3 - 14 x 2 + 8x + 15 x 2 - 21x + 12 Simplify



-4 x 3 - 8 x 2 + 3x + 24 x 2 + 48x - 18 Simplify


-21 x 3 + 31 x 2 + 2x - 8

x

3x

28 x 2

-21 x 3

31 x 2



A2_MNLESE385894_U3M06L2 285 6815 1040 PM

1 2 3






multiplied








x 2 -x +1

x 2 x 4 -x 3 + x 2

+3x +3 x 3 -3 x 2 +3x

-5 -5 x 2 +5x -5


285 Lesson 6 2


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002 x 2 + 02x + 3

times - 0002 x 2 - 01x + 2

――― 004 x 2 + 04x + 6

-0002 x 3 - 002 x 2 - 03x -000004 x 4 - 00004 x 3 - 0006 x 2

――――― -000004 x 4 - 00024 x 3 + 0014 x 2 + 01x + 6


T (x) = -000004 x 4 - 00024 x 3 + 0014 x 2 + 01x + 6





03 x 2 - 16x + 14

times - 0001 x 2 + 83

―――― 249 x 2 - 1328x

-0018 x 3 - 084x

-00003 x + 00016 x 3 - 0014 x 2

――――――― -00003 x - 00164 x 3 + 2572 x 2 + 1162


T (x) =

-006x

+1162

-00003 x 4 - 00164 x 3 + 2572 x 2 - 1412x + 1162

4

4

+0096 x 2

-1412x







exponents





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Your Turn



(x + a) 2 - (x - a) 2 = 4ax



(x + a) 2 - (x - a) 2 = 4ax


(x + a) 2 - (x - a) 2 = 4ax



4ax = 4ax Simplify




07 x 2 - 2x + 23

times - 0004 x 2 - 008x + 25

――――― 175 x 2 - 50x + 575

-0056 x 3 + 016 x 2 - 184x

-00028 x 4 +thinsp 0008 x 3 - 0092 x 2

―――――― -00028 x 4 - 0048 x 3 + 17568 x 2 - 5184x + 575



T (x) = -00028 x 4 - 0048 x 3 + 17568 x 2 - 5184x + 575



A2_MNLESE385894_U3M06L2 287 61615 839 AM






LANGUAGE SUPPORT








287 Lesson 6 2


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(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


a 3 - a 2 b + a b 2 + - a b 2 + = a 3 + b 3




Your Turn







( 4 2 + 3 2 ) 2 = ( 4 2 - 3 2 ) 2 + (2 middot 4 middot 3) 2 (16 + 9) 2 = (16 - 9) 2 + (24) 2 (25) 2 = (7) 2 + (24) 2 625 = 49 + 576

625 = 625





-ab

a 2 b

a 2 b Simplify

true

a b 2

b 3

+

b




A2_MNLESE385894_U3M06L2 288 62714 118 PM






calculations



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Let x = and y = 7

Evaluate

(20 + ) 2

= 2 0 2 + + 7 2

2 7 2 = 400 + + 49

2 7 2 =


2 7 2 =

Your Turn





20


280

729

729


(40 + 3) (40 - 3) = 40 2 - 3 2

43 37 = 1600 - 9

43 37 = 1591



(20 - 2) 2 = 20 2 - 2 2 20 + 2 2

= 400 - 80 + 4

= 324



A2_MNLESE385894_U3M06L2 289 161014 838 AM


289 Lesson 6 2


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Elaborate








2 (3x - 2) (2 x 2 + 3x - 1)

3 ( x 3 + 3 x 2 + 1) (3 x 2 + 6x - 2)



x 3 +3 x 2 +1

times 3 x 2 +6x -2






desired unit


A polynomial

(x + 5) (x + 1) x = x 3 + 6 x 2 + 5x


6 x 3 + 9 x 2 - 3x - 4 x 2 - 6x +2 Simplify
















property applies






EVALUATE

ASSIGNMENT GUIDE



Exercise 1


Exercises 2ndash7


Exercises 8ndash11


Exercises 12ndash15


Exercises 16ndash21



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4 ( x 2 + 9x + 7) (3 x 2 + 9x + 5)

5 (2x + 5y) (3 x 2 - 4xy + 2 y 2 )

6 ( x 3 + x 2 + 1) ( x 2 - x - 5)

7 (4 x 2 + 3x + 2) (3 x 2 + 2x - 1)

x 2 +9x +7

times 3 x 2 +9x +5








x 3 + x 2 +1

times x 2 -x -5





4 x 2 + 3x +2

times 3 x 2 + 2x -1













291 Lesson 6 2


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R (x) = (02x + 5) (-01 x 3 + x 2 - 3x + 4)

= 02x (-01 x 3 ) + 02x ( x 2 ) + 02x (-3x) + 02x (4) + 5 (-01 x 3 ) + 5 ( x 2 ) + 5 (-3x) + 5 (4)

= -002 x 4 + 02 x 3 - 0 6x 2 + 08x - 05x 3 + 5x 2 - 15x + 20

=-002 x 4 - 0 3x 3 + 44 x 2 - 142x + 20


V (x) = ( x 5 - 3 x 4 + 10 x 2 - 6x + 30) (0004 x 4 - 002 x 2 + 03x + 4)

x 5 -3 x 4 +10 x 2 -6x +30


03 x 6 -09 x 5 +3 x 3 -18 x 2 +9x

-002 x 7 +006 x 6 -02 x 4 +012 x 3 -06 x 2

0004 x 9 -0012 x 8 +004 x 6 -0024 x 5 +012 x 4

――――――――――― 0004 x 9 -0012 x 8 -002 x 7 +04 x 6 +3076 x 5 -1208 x 4 +312 x 3 +376 x 2 -15x + 120

V (x) = 0004 x 4 - 0012 x 8 - 002 x 7 + 04 x 6 + 3076 x 5 - 1208 x 4 + 312 x 3 + 376 x 2 -15x + 120

T (y) = (4 y 2 + y) (2 y 3 + 3 y 2 + y)

= 8 y 5 + 12 y 4 + 4 y 3 + 2 y 4 + 3 y 3 + y 2

= 8 y 5 + 14 y 4 + 7 y 3 + y 2




A2_MNLESE385894_U3M06L2 292 61615 840 AM




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12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)






= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz


a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5


x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4





x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









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a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

X

X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


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2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









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002 x 2 + 02x + 3

times - 0002 x 2 - 01x + 2

――― 004 x 2 + 04x + 6

-0002 x 3 - 002 x 2 - 03x -000004 x 4 - 00004 x 3 - 0006 x 2

――――― -000004 x 4 - 00024 x 3 + 0014 x 2 + 01x + 6


T (x) = -000004 x 4 - 00024 x 3 + 0014 x 2 + 01x + 6





03 x 2 - 16x + 14

times - 0001 x 2 + 83

―――― 249 x 2 - 1328x

-0018 x 3 - 084x

-00003 x + 00016 x 3 - 0014 x 2

――――――― -00003 x - 00164 x 3 + 2572 x 2 + 1162


T (x) =

-006x

+1162

-00003 x 4 - 00164 x 3 + 2572 x 2 - 1412x + 1162

4

4

+0096 x 2

-1412x







exponents





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Your Turn



(x + a) 2 - (x - a) 2 = 4ax



(x + a) 2 - (x - a) 2 = 4ax


(x + a) 2 - (x - a) 2 = 4ax



4ax = 4ax Simplify




07 x 2 - 2x + 23

times - 0004 x 2 - 008x + 25

――――― 175 x 2 - 50x + 575

-0056 x 3 + 016 x 2 - 184x

-00028 x 4 +thinsp 0008 x 3 - 0092 x 2

―――――― -00028 x 4 - 0048 x 3 + 17568 x 2 - 5184x + 575



T (x) = -00028 x 4 - 0048 x 3 + 17568 x 2 - 5184x + 575



A2_MNLESE385894_U3M06L2 287 61615 839 AM






LANGUAGE SUPPORT








287 Lesson 6 2


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(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


a 3 - a 2 b + a b 2 + - a b 2 + = a 3 + b 3




Your Turn







( 4 2 + 3 2 ) 2 = ( 4 2 - 3 2 ) 2 + (2 middot 4 middot 3) 2 (16 + 9) 2 = (16 - 9) 2 + (24) 2 (25) 2 = (7) 2 + (24) 2 625 = 49 + 576

625 = 625





-ab

a 2 b

a 2 b Simplify

true

a b 2

b 3

+

b




A2_MNLESE385894_U3M06L2 288 62714 118 PM






calculations



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Let x = and y = 7

Evaluate

(20 + ) 2

= 2 0 2 + + 7 2

2 7 2 = 400 + + 49

2 7 2 =


2 7 2 =

Your Turn





20


280

729

729


(40 + 3) (40 - 3) = 40 2 - 3 2

43 37 = 1600 - 9

43 37 = 1591



(20 - 2) 2 = 20 2 - 2 2 20 + 2 2

= 400 - 80 + 4

= 324



A2_MNLESE385894_U3M06L2 289 161014 838 AM


289 Lesson 6 2


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Elaborate








2 (3x - 2) (2 x 2 + 3x - 1)

3 ( x 3 + 3 x 2 + 1) (3 x 2 + 6x - 2)



x 3 +3 x 2 +1

times 3 x 2 +6x -2






desired unit


A polynomial

(x + 5) (x + 1) x = x 3 + 6 x 2 + 5x


6 x 3 + 9 x 2 - 3x - 4 x 2 - 6x +2 Simplify
















property applies






EVALUATE

ASSIGNMENT GUIDE



Exercise 1


Exercises 2ndash7


Exercises 8ndash11


Exercises 12ndash15


Exercises 16ndash21



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4 ( x 2 + 9x + 7) (3 x 2 + 9x + 5)

5 (2x + 5y) (3 x 2 - 4xy + 2 y 2 )

6 ( x 3 + x 2 + 1) ( x 2 - x - 5)

7 (4 x 2 + 3x + 2) (3 x 2 + 2x - 1)

x 2 +9x +7

times 3 x 2 +9x +5








x 3 + x 2 +1

times x 2 -x -5





4 x 2 + 3x +2

times 3 x 2 + 2x -1













291 Lesson 6 2


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R (x) = (02x + 5) (-01 x 3 + x 2 - 3x + 4)

= 02x (-01 x 3 ) + 02x ( x 2 ) + 02x (-3x) + 02x (4) + 5 (-01 x 3 ) + 5 ( x 2 ) + 5 (-3x) + 5 (4)

= -002 x 4 + 02 x 3 - 0 6x 2 + 08x - 05x 3 + 5x 2 - 15x + 20

=-002 x 4 - 0 3x 3 + 44 x 2 - 142x + 20


V (x) = ( x 5 - 3 x 4 + 10 x 2 - 6x + 30) (0004 x 4 - 002 x 2 + 03x + 4)

x 5 -3 x 4 +10 x 2 -6x +30


03 x 6 -09 x 5 +3 x 3 -18 x 2 +9x

-002 x 7 +006 x 6 -02 x 4 +012 x 3 -06 x 2

0004 x 9 -0012 x 8 +004 x 6 -0024 x 5 +012 x 4

――――――――――― 0004 x 9 -0012 x 8 -002 x 7 +04 x 6 +3076 x 5 -1208 x 4 +312 x 3 +376 x 2 -15x + 120

V (x) = 0004 x 4 - 0012 x 8 - 002 x 7 + 04 x 6 + 3076 x 5 - 1208 x 4 + 312 x 3 + 376 x 2 -15x + 120

T (y) = (4 y 2 + y) (2 y 3 + 3 y 2 + y)

= 8 y 5 + 12 y 4 + 4 y 3 + 2 y 4 + 3 y 3 + y 2

= 8 y 5 + 14 y 4 + 7 y 3 + y 2




A2_MNLESE385894_U3M06L2 292 61615 840 AM




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12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)






= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz


a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5


x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4





x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









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a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

X

X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


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2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









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Your Turn



(x + a) 2 - (x - a) 2 = 4ax



(x + a) 2 - (x - a) 2 = 4ax


(x + a) 2 - (x - a) 2 = 4ax



4ax = 4ax Simplify




07 x 2 - 2x + 23

times - 0004 x 2 - 008x + 25

――――― 175 x 2 - 50x + 575

-0056 x 3 + 016 x 2 - 184x

-00028 x 4 +thinsp 0008 x 3 - 0092 x 2

―――――― -00028 x 4 - 0048 x 3 + 17568 x 2 - 5184x + 575



T (x) = -00028 x 4 - 0048 x 3 + 17568 x 2 - 5184x + 575



A2_MNLESE385894_U3M06L2 287 61615 839 AM






LANGUAGE SUPPORT








287 Lesson 6 2


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(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


a 3 - a 2 b + a b 2 + - a b 2 + = a 3 + b 3




Your Turn







( 4 2 + 3 2 ) 2 = ( 4 2 - 3 2 ) 2 + (2 middot 4 middot 3) 2 (16 + 9) 2 = (16 - 9) 2 + (24) 2 (25) 2 = (7) 2 + (24) 2 625 = 49 + 576

625 = 625





-ab

a 2 b

a 2 b Simplify

true

a b 2

b 3

+

b




A2_MNLESE385894_U3M06L2 288 62714 118 PM






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Let x = and y = 7

Evaluate

(20 + ) 2

= 2 0 2 + + 7 2

2 7 2 = 400 + + 49

2 7 2 =


2 7 2 =

Your Turn





20


280

729

729


(40 + 3) (40 - 3) = 40 2 - 3 2

43 37 = 1600 - 9

43 37 = 1591



(20 - 2) 2 = 20 2 - 2 2 20 + 2 2

= 400 - 80 + 4

= 324



A2_MNLESE385894_U3M06L2 289 161014 838 AM


289 Lesson 6 2


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Elaborate








2 (3x - 2) (2 x 2 + 3x - 1)

3 ( x 3 + 3 x 2 + 1) (3 x 2 + 6x - 2)



x 3 +3 x 2 +1

times 3 x 2 +6x -2






desired unit


A polynomial

(x + 5) (x + 1) x = x 3 + 6 x 2 + 5x


6 x 3 + 9 x 2 - 3x - 4 x 2 - 6x +2 Simplify
















property applies






EVALUATE

ASSIGNMENT GUIDE



Exercise 1


Exercises 2ndash7


Exercises 8ndash11


Exercises 12ndash15


Exercises 16ndash21



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4 ( x 2 + 9x + 7) (3 x 2 + 9x + 5)

5 (2x + 5y) (3 x 2 - 4xy + 2 y 2 )

6 ( x 3 + x 2 + 1) ( x 2 - x - 5)

7 (4 x 2 + 3x + 2) (3 x 2 + 2x - 1)

x 2 +9x +7

times 3 x 2 +9x +5








x 3 + x 2 +1

times x 2 -x -5





4 x 2 + 3x +2

times 3 x 2 + 2x -1













291 Lesson 6 2


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R (x) = (02x + 5) (-01 x 3 + x 2 - 3x + 4)

= 02x (-01 x 3 ) + 02x ( x 2 ) + 02x (-3x) + 02x (4) + 5 (-01 x 3 ) + 5 ( x 2 ) + 5 (-3x) + 5 (4)

= -002 x 4 + 02 x 3 - 0 6x 2 + 08x - 05x 3 + 5x 2 - 15x + 20

=-002 x 4 - 0 3x 3 + 44 x 2 - 142x + 20


V (x) = ( x 5 - 3 x 4 + 10 x 2 - 6x + 30) (0004 x 4 - 002 x 2 + 03x + 4)

x 5 -3 x 4 +10 x 2 -6x +30


03 x 6 -09 x 5 +3 x 3 -18 x 2 +9x

-002 x 7 +006 x 6 -02 x 4 +012 x 3 -06 x 2

0004 x 9 -0012 x 8 +004 x 6 -0024 x 5 +012 x 4

――――――――――― 0004 x 9 -0012 x 8 -002 x 7 +04 x 6 +3076 x 5 -1208 x 4 +312 x 3 +376 x 2 -15x + 120

V (x) = 0004 x 4 - 0012 x 8 - 002 x 7 + 04 x 6 + 3076 x 5 - 1208 x 4 + 312 x 3 + 376 x 2 -15x + 120

T (y) = (4 y 2 + y) (2 y 3 + 3 y 2 + y)

= 8 y 5 + 12 y 4 + 4 y 3 + 2 y 4 + 3 y 3 + y 2

= 8 y 5 + 14 y 4 + 7 y 3 + y 2




A2_MNLESE385894_U3M06L2 292 61615 840 AM




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12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)






= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz


a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5


x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4





x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









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a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

X

X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


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2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









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(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


(a + b) ( a 2 - ab + b 2 ) = a 3 + b 3


a 3 - a 2 b + a b 2 + - a b 2 + = a 3 + b 3




Your Turn







( 4 2 + 3 2 ) 2 = ( 4 2 - 3 2 ) 2 + (2 middot 4 middot 3) 2 (16 + 9) 2 = (16 - 9) 2 + (24) 2 (25) 2 = (7) 2 + (24) 2 625 = 49 + 576

625 = 625





-ab

a 2 b

a 2 b Simplify

true

a b 2

b 3

+

b




A2_MNLESE385894_U3M06L2 288 62714 118 PM






calculations



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Let x = and y = 7

Evaluate

(20 + ) 2

= 2 0 2 + + 7 2

2 7 2 = 400 + + 49

2 7 2 =


2 7 2 =

Your Turn





20


280

729

729


(40 + 3) (40 - 3) = 40 2 - 3 2

43 37 = 1600 - 9

43 37 = 1591



(20 - 2) 2 = 20 2 - 2 2 20 + 2 2

= 400 - 80 + 4

= 324



A2_MNLESE385894_U3M06L2 289 161014 838 AM


289 Lesson 6 2


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Elaborate








2 (3x - 2) (2 x 2 + 3x - 1)

3 ( x 3 + 3 x 2 + 1) (3 x 2 + 6x - 2)



x 3 +3 x 2 +1

times 3 x 2 +6x -2






desired unit


A polynomial

(x + 5) (x + 1) x = x 3 + 6 x 2 + 5x


6 x 3 + 9 x 2 - 3x - 4 x 2 - 6x +2 Simplify
















property applies






EVALUATE

ASSIGNMENT GUIDE



Exercise 1


Exercises 2ndash7


Exercises 8ndash11


Exercises 12ndash15


Exercises 16ndash21



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4 ( x 2 + 9x + 7) (3 x 2 + 9x + 5)

5 (2x + 5y) (3 x 2 - 4xy + 2 y 2 )

6 ( x 3 + x 2 + 1) ( x 2 - x - 5)

7 (4 x 2 + 3x + 2) (3 x 2 + 2x - 1)

x 2 +9x +7

times 3 x 2 +9x +5








x 3 + x 2 +1

times x 2 -x -5





4 x 2 + 3x +2

times 3 x 2 + 2x -1













291 Lesson 6 2


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R (x) = (02x + 5) (-01 x 3 + x 2 - 3x + 4)

= 02x (-01 x 3 ) + 02x ( x 2 ) + 02x (-3x) + 02x (4) + 5 (-01 x 3 ) + 5 ( x 2 ) + 5 (-3x) + 5 (4)

= -002 x 4 + 02 x 3 - 0 6x 2 + 08x - 05x 3 + 5x 2 - 15x + 20

=-002 x 4 - 0 3x 3 + 44 x 2 - 142x + 20


V (x) = ( x 5 - 3 x 4 + 10 x 2 - 6x + 30) (0004 x 4 - 002 x 2 + 03x + 4)

x 5 -3 x 4 +10 x 2 -6x +30


03 x 6 -09 x 5 +3 x 3 -18 x 2 +9x

-002 x 7 +006 x 6 -02 x 4 +012 x 3 -06 x 2

0004 x 9 -0012 x 8 +004 x 6 -0024 x 5 +012 x 4

――――――――――― 0004 x 9 -0012 x 8 -002 x 7 +04 x 6 +3076 x 5 -1208 x 4 +312 x 3 +376 x 2 -15x + 120

V (x) = 0004 x 4 - 0012 x 8 - 002 x 7 + 04 x 6 + 3076 x 5 - 1208 x 4 + 312 x 3 + 376 x 2 -15x + 120

T (y) = (4 y 2 + y) (2 y 3 + 3 y 2 + y)

= 8 y 5 + 12 y 4 + 4 y 3 + 2 y 4 + 3 y 3 + y 2

= 8 y 5 + 14 y 4 + 7 y 3 + y 2




A2_MNLESE385894_U3M06L2 292 61615 840 AM




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12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)






= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz


a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5


x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4





x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









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a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

X

X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


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2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









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Let x = and y = 7

Evaluate

(20 + ) 2

= 2 0 2 + + 7 2

2 7 2 = 400 + + 49

2 7 2 =


2 7 2 =

Your Turn





20


280

729

729


(40 + 3) (40 - 3) = 40 2 - 3 2

43 37 = 1600 - 9

43 37 = 1591



(20 - 2) 2 = 20 2 - 2 2 20 + 2 2

= 400 - 80 + 4

= 324



A2_MNLESE385894_U3M06L2 289 161014 838 AM


289 Lesson 6 2


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Elaborate








2 (3x - 2) (2 x 2 + 3x - 1)

3 ( x 3 + 3 x 2 + 1) (3 x 2 + 6x - 2)



x 3 +3 x 2 +1

times 3 x 2 +6x -2






desired unit


A polynomial

(x + 5) (x + 1) x = x 3 + 6 x 2 + 5x


6 x 3 + 9 x 2 - 3x - 4 x 2 - 6x +2 Simplify
















property applies






EVALUATE

ASSIGNMENT GUIDE



Exercise 1


Exercises 2ndash7


Exercises 8ndash11


Exercises 12ndash15


Exercises 16ndash21



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4 ( x 2 + 9x + 7) (3 x 2 + 9x + 5)

5 (2x + 5y) (3 x 2 - 4xy + 2 y 2 )

6 ( x 3 + x 2 + 1) ( x 2 - x - 5)

7 (4 x 2 + 3x + 2) (3 x 2 + 2x - 1)

x 2 +9x +7

times 3 x 2 +9x +5








x 3 + x 2 +1

times x 2 -x -5





4 x 2 + 3x +2

times 3 x 2 + 2x -1













291 Lesson 6 2


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R (x) = (02x + 5) (-01 x 3 + x 2 - 3x + 4)

= 02x (-01 x 3 ) + 02x ( x 2 ) + 02x (-3x) + 02x (4) + 5 (-01 x 3 ) + 5 ( x 2 ) + 5 (-3x) + 5 (4)

= -002 x 4 + 02 x 3 - 0 6x 2 + 08x - 05x 3 + 5x 2 - 15x + 20

=-002 x 4 - 0 3x 3 + 44 x 2 - 142x + 20


V (x) = ( x 5 - 3 x 4 + 10 x 2 - 6x + 30) (0004 x 4 - 002 x 2 + 03x + 4)

x 5 -3 x 4 +10 x 2 -6x +30


03 x 6 -09 x 5 +3 x 3 -18 x 2 +9x

-002 x 7 +006 x 6 -02 x 4 +012 x 3 -06 x 2

0004 x 9 -0012 x 8 +004 x 6 -0024 x 5 +012 x 4

――――――――――― 0004 x 9 -0012 x 8 -002 x 7 +04 x 6 +3076 x 5 -1208 x 4 +312 x 3 +376 x 2 -15x + 120

V (x) = 0004 x 4 - 0012 x 8 - 002 x 7 + 04 x 6 + 3076 x 5 - 1208 x 4 + 312 x 3 + 376 x 2 -15x + 120

T (y) = (4 y 2 + y) (2 y 3 + 3 y 2 + y)

= 8 y 5 + 12 y 4 + 4 y 3 + 2 y 4 + 3 y 3 + y 2

= 8 y 5 + 14 y 4 + 7 y 3 + y 2




A2_MNLESE385894_U3M06L2 292 61615 840 AM




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12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)






= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz


a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5


x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4





x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









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a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

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A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


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2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









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Elaborate








2 (3x - 2) (2 x 2 + 3x - 1)

3 ( x 3 + 3 x 2 + 1) (3 x 2 + 6x - 2)



x 3 +3 x 2 +1

times 3 x 2 +6x -2






desired unit


A polynomial

(x + 5) (x + 1) x = x 3 + 6 x 2 + 5x


6 x 3 + 9 x 2 - 3x - 4 x 2 - 6x +2 Simplify
















property applies






EVALUATE

ASSIGNMENT GUIDE



Exercise 1


Exercises 2ndash7


Exercises 8ndash11


Exercises 12ndash15


Exercises 16ndash21



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4 ( x 2 + 9x + 7) (3 x 2 + 9x + 5)

5 (2x + 5y) (3 x 2 - 4xy + 2 y 2 )

6 ( x 3 + x 2 + 1) ( x 2 - x - 5)

7 (4 x 2 + 3x + 2) (3 x 2 + 2x - 1)

x 2 +9x +7

times 3 x 2 +9x +5








x 3 + x 2 +1

times x 2 -x -5





4 x 2 + 3x +2

times 3 x 2 + 2x -1













291 Lesson 6 2


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R (x) = (02x + 5) (-01 x 3 + x 2 - 3x + 4)

= 02x (-01 x 3 ) + 02x ( x 2 ) + 02x (-3x) + 02x (4) + 5 (-01 x 3 ) + 5 ( x 2 ) + 5 (-3x) + 5 (4)

= -002 x 4 + 02 x 3 - 0 6x 2 + 08x - 05x 3 + 5x 2 - 15x + 20

=-002 x 4 - 0 3x 3 + 44 x 2 - 142x + 20


V (x) = ( x 5 - 3 x 4 + 10 x 2 - 6x + 30) (0004 x 4 - 002 x 2 + 03x + 4)

x 5 -3 x 4 +10 x 2 -6x +30


03 x 6 -09 x 5 +3 x 3 -18 x 2 +9x

-002 x 7 +006 x 6 -02 x 4 +012 x 3 -06 x 2

0004 x 9 -0012 x 8 +004 x 6 -0024 x 5 +012 x 4

――――――――――― 0004 x 9 -0012 x 8 -002 x 7 +04 x 6 +3076 x 5 -1208 x 4 +312 x 3 +376 x 2 -15x + 120

V (x) = 0004 x 4 - 0012 x 8 - 002 x 7 + 04 x 6 + 3076 x 5 - 1208 x 4 + 312 x 3 + 376 x 2 -15x + 120

T (y) = (4 y 2 + y) (2 y 3 + 3 y 2 + y)

= 8 y 5 + 12 y 4 + 4 y 3 + 2 y 4 + 3 y 3 + y 2

= 8 y 5 + 14 y 4 + 7 y 3 + y 2




A2_MNLESE385894_U3M06L2 292 61615 840 AM




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12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)






= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz


a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5


x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4





x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









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a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

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X

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A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


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2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









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4 ( x 2 + 9x + 7) (3 x 2 + 9x + 5)

5 (2x + 5y) (3 x 2 - 4xy + 2 y 2 )

6 ( x 3 + x 2 + 1) ( x 2 - x - 5)

7 (4 x 2 + 3x + 2) (3 x 2 + 2x - 1)

x 2 +9x +7

times 3 x 2 +9x +5








x 3 + x 2 +1

times x 2 -x -5





4 x 2 + 3x +2

times 3 x 2 + 2x -1













291 Lesson 6 2


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R (x) = (02x + 5) (-01 x 3 + x 2 - 3x + 4)

= 02x (-01 x 3 ) + 02x ( x 2 ) + 02x (-3x) + 02x (4) + 5 (-01 x 3 ) + 5 ( x 2 ) + 5 (-3x) + 5 (4)

= -002 x 4 + 02 x 3 - 0 6x 2 + 08x - 05x 3 + 5x 2 - 15x + 20

=-002 x 4 - 0 3x 3 + 44 x 2 - 142x + 20


V (x) = ( x 5 - 3 x 4 + 10 x 2 - 6x + 30) (0004 x 4 - 002 x 2 + 03x + 4)

x 5 -3 x 4 +10 x 2 -6x +30


03 x 6 -09 x 5 +3 x 3 -18 x 2 +9x

-002 x 7 +006 x 6 -02 x 4 +012 x 3 -06 x 2

0004 x 9 -0012 x 8 +004 x 6 -0024 x 5 +012 x 4

――――――――――― 0004 x 9 -0012 x 8 -002 x 7 +04 x 6 +3076 x 5 -1208 x 4 +312 x 3 +376 x 2 -15x + 120

V (x) = 0004 x 4 - 0012 x 8 - 002 x 7 + 04 x 6 + 3076 x 5 - 1208 x 4 + 312 x 3 + 376 x 2 -15x + 120

T (y) = (4 y 2 + y) (2 y 3 + 3 y 2 + y)

= 8 y 5 + 12 y 4 + 4 y 3 + 2 y 4 + 3 y 3 + y 2

= 8 y 5 + 14 y 4 + 7 y 3 + y 2




A2_MNLESE385894_U3M06L2 292 61615 840 AM




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12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)






= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz


a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5


x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4





x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









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a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

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X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


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2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









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R (x) = (02x + 5) (-01 x 3 + x 2 - 3x + 4)

= 02x (-01 x 3 ) + 02x ( x 2 ) + 02x (-3x) + 02x (4) + 5 (-01 x 3 ) + 5 ( x 2 ) + 5 (-3x) + 5 (4)

= -002 x 4 + 02 x 3 - 0 6x 2 + 08x - 05x 3 + 5x 2 - 15x + 20

=-002 x 4 - 0 3x 3 + 44 x 2 - 142x + 20


V (x) = ( x 5 - 3 x 4 + 10 x 2 - 6x + 30) (0004 x 4 - 002 x 2 + 03x + 4)

x 5 -3 x 4 +10 x 2 -6x +30


03 x 6 -09 x 5 +3 x 3 -18 x 2 +9x

-002 x 7 +006 x 6 -02 x 4 +012 x 3 -06 x 2

0004 x 9 -0012 x 8 +004 x 6 -0024 x 5 +012 x 4

――――――――――― 0004 x 9 -0012 x 8 -002 x 7 +04 x 6 +3076 x 5 -1208 x 4 +312 x 3 +376 x 2 -15x + 120

V (x) = 0004 x 4 - 0012 x 8 - 002 x 7 + 04 x 6 + 3076 x 5 - 1208 x 4 + 312 x 3 + 376 x 2 -15x + 120

T (y) = (4 y 2 + y) (2 y 3 + 3 y 2 + y)

= 8 y 5 + 12 y 4 + 4 y 3 + 2 y 4 + 3 y 3 + y 2

= 8 y 5 + 14 y 4 + 7 y 3 + y 2




A2_MNLESE385894_U3M06L2 292 61615 840 AM




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12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)






= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz


a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5


x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4





x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









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a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

X

X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


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2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









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12 (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz

13 a 5 + b 5 = (a + b) ( a 4 - a 3 b + a 2 b 2 - a b 3 + b 4 )

14 x 4 - y 4 = (x - y) (x + y) ( x 2 + y 2 )

K = 1 _ 2 (2) (-98t + 24) 2

K = (-98t + 24) 2

K = 9604 t 2 - 4704t + 576

U = 98 (2) (-49 t 2 + 24t + 60)

U = -9604 t 2 + 4704t + 1176

K + U = (9604 t 2 - 4704t + 576) + (-9604 t 2 + 4704t + 1176)






= x 2 + y 2 + z 2 + 2xy + 2xz + 2yz


a 4 - a 3 b + a 2 b 2 - a b 3 + b 4

times a + b

―――― a 4 b - a 3 b 2 + a 2 b 3 - a b 4 + b 5

a 5 - a 4 b + a 3 b 2 - a 2 b 3 + a b 4

―――― a 5 + b 5


x 4 - y 4 = ( x 2 - y 2 ) ( x 2 + y 2 )

x 4 - y 4 = x 2 ( x 2 ) + x 2 ( y 2 ) - y 2 ( x 2 ) - y 2 ( y 2 )

x 4 - y 4 = x 4 - y 4





x 3 3x ndash2

x 2

2x

1

293 Lesson 6 2


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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









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a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

X

X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


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2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









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15 ( a 2 + b 2 ) ( x 2 + y 2 ) = (ax - by) 2 + (bx + ay) 2



16 4 3 2 17 3 2 2

18 8 9 2 19 4 7 2

20 54 sdot 38 21 58 sdot 68




2 + 2 (bx) (ay) + (ay) 2



4 3 2 = (40 + 3) 2

= 1600 + 240 + 9

= 1849

3 2 2 = (30 + 2) 2

= 900 + 120 + 4

= 1024

8 9 2 = (90 - 1) 2

= 8100 - 180 + 1

= 7921

54 sdot 38 = (46 + 8) (46 - 8)

= 4 6 2 - 8 2

= (50 - 4) 2 - 64

= 5 0 2 - 2 sdot 50 sdot 4 + 4 2 - 64

= 2500 - 400 + 16 - 64

= 2052



64 - 27x6 = 43 - (3 x 2 ) 3

= (4 - 3x2) (42 + 4 (3x2) + (3x2) 2)

= (4 - 3x2) (16 + 12x2 + 9x4)

58 sdot 68 = (63 + 5) (63 - 5)

= 6 3 2 - 5 2

= (60 + 3) 2 - 25

= 6 0 2 + 2 sdot 60 sdot 3 + 3 2 - 25

= 3600 + 360 + 9 - 25

= 3944

4 7 2 = (50 - 3) 2

= 2500 - 300 + 9

= 2209









copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany


a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

X

X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany


a (5x) (3x) 1 2 3 4

b (3x) (2x + 1) 1 2 3 4

c (x + 1) (x - 1) 1 2 3 4

d (x + 2) ( 3x 2 - 2x + 1) 1 2 3 4







x 6 + y 6 = ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )


x 6 + y 6 = x 6 - x 4 y 2 + x 2 y 4 + x 4 y 2 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + x 4 y 2 - x 4 y 2 + x 2 y 4 - x 2 y 4 + y 6

x 6 + y 6 = x 6 + y 6


3 + ( y 2 )

3


simplify ( x 2 ) 3 + ( y 2 )

3

( x 2 ) 3 + ( y 2 )

3 = ( x 2 + y 2 ) ( ( x 2 )

2 - ( x 2 ) ( y 2 ) + ( y 2 )

2 )

= ( x 2 + y 2 ) ( x 4 - x 2 y 2 + y 4 )



X

X

X

X



A2_MNLESE385894_U3M06L2 295 100615 1249 AM



295 Lesson 6 2


copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









copy H

oug

hton Mifflin H

arcourt Publishin

g Com

pany




2008 83443 2178

2007 82832 2053

2006 82284 2108

2005 82551 2006

2004 83222 210

2003 83415 212

2002 83730 216

2001 84160 224

2000 84432 224

1999 85043 229

1998 85691 249

1997 86765 262

1996 88144 266

1995 90557 265

1994 91289 273

1993 92377 287

1992 93192 299






b 0884 t 3 +396 t 2 -1510t +93700

times 000418 t 2 -0125t +301

_________

266084 t 3 +119196 t 2 -45451t +282037

-01105 t 4 -495 t 3 +18875 t 2 -117125t

000369512 t 5 + 0165528 t 4 -63118 t 3 +391666 t 2

――――――――― 000369512 t 5 + 0055028 t 4 -860096 t 3 +699612 t 2 -162576t +282037


D (t) = 000370 t 5 + 00550 t 4 - 860 t 3 + 700 t 2 - 16300t + 282000




EXTENSION ACTIVITY









Multiplying Polynomials

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Transcript of Multiplying Polynomials