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IntroductionWe have studied about a particular type of algebraic expression, called polynomial, and the terminology related to it. Now, we shall study the factorisation of polynomials. In addition, we shall study some more algebraic identities and their use in factorisation.

Polynomials in one variable: A polynomial p(x) in one variable x is an algebraic expression in x of the form p(x) = anx

n + an – 1xn – 1 + ... + a2x

2 + a1x + a0, where a0, a1, a2, ...., an are constants and an ≠ 0. Here, a0, a1, a2, ..., an are respectively the coefficients of x0, x, x2, ..., xn, and n (the highest power is called the degree of the polynomial p(x). Each of anx

n, an – 1xn – 1, ..., a0 with an ≠ 0, is called a term of

polynomial (px).

Polynomials are classified according to the number of their terms as well as according to their degree.

(i) A polynomial of one term is called a monomial.

Examples: 5x2, 4x, –54x3, etc.

(ii) A polynomial of two terms is called a binomial.

Examples: x + 1, x2 – x, y2 + 1, etc.

(iii) A polynomial of three terms called a trinomial.

Examples: x2 + x + 2, 2 2+ −x x , etc.

(iv) A polynomial of degree 0 is called a constant polynomial.

Examples: 12, 74, –84, etc.

(v) A polynomial of degree 1 is called a linear polynomial.

Examples: 2x – 2, 2 1y + , etc.

(vi) A polynomial of degree 2 is called a quadratic polynomial.

Examples: 2x2 + 5, 5x2 + 3x, etc.

(vii) A polynomial of degree 3 is called a cubic polynomial.

Examples: 3x3, 2x3 + 1, 5x3 + x2, etc.

Algebraic identities: An algebraic identity is an algebraic equation that is true for all values of the variables occurring in it. Algebraic identities are used to factorise algebraic expressions and also in computations.

Some algebraic identities are as follows:

(i) (a + b)2 = a2 + 2ab + b2 (ii) (a – b)2 = a2 – 2ab + b2

(iii) a2 – b2 = (a + b)(a – b) (iv) (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

(v) (a + b)3 = a3 + b3 + 3a2b + 3ab2 (vi) (a – b)3 = a3 – b3 – 3ab(a – b)

(vii) a3 + b3 = (a + b)(a2 – ab + b2) (viii) a3 – b3 = (a – b)(a2 + ab + b2)

Multiplication of polynomials: To multiply two polynomials, multiply each term in one polynomial by each term in the other polynomial and add those answers together. Simplify if needed.

Factorisation of polynomials: To factorise quadratic polynomials of the type ax2 + bx + c, where a ≠ 0 and a, b, c are constants, we have to write b as the sum of two numbers whose product is ac.

Algebra2

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Activity 2.1 Algebraic Identity : (a + b)3 = a3 + 3ab(a + b) + b3

To compute (a + b)3, we extend the identity (a + b)2 = a2 + 2ab + b2 as follows: (a + b)3 = (a + b)(a + b)2 = (a + b)(a2 + 2ab + b2) = a(a2 + 2ab + b2) + b(a2 + 2ab + b 2) = a3 + 2a2b + ab2 + a2b + 2ab2 + b3

= a3 + 3a2b + 3ab2 + b3 = a3 + 3ab(a + b) + b3

ObjectiveTo verify the algebraic identity: (a + b)3 = a 3 + 3ab(a + b) + b 3.

Pre-requisite knowledge (i ) Concept of a cube and a cuboid (ii ) Volume of a cube = Side × Side × Side (iii ) Volume of a cuboid = Length × Breadth × Height

Procedure (i ) Make a cube of side ‘a’ units as shown in the figure. Its volume is a3. Wrap a red glazed paper on it.

Fig. 1 (ii ) Make another cube of side ‘b’ units as shown in the figure. Its volume is b3. Wrap a black glazed

paper on it.

Fig. 2 (iii ) Make three cuboids each of dimensions (a) × (b) × (a + b) units. Wrap them with green glazed papers

as shown in the figure. Volume of each cuboid is a · b(a + b) cubic units.

Fig. 3

Materials Required Cardboard Glazed paper of various colours Sheets of white paper A pair of scissors Scale,Cutter,Sketchpens Geometry box, Fevicol

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(iv ) Arrange the above two cubes and three cuboids in such a way that they altogether make a cube as shown in the figure.

Fig. 4 (v ) A cubed binomial (sum) is equal to the cube of the first, plus three times the square of the first by

the second, plus three times the first by the square of the second, plus the cube of the second.

Observations (i ) We have joined the above five blocks (two cubes and three cuboids) to form a cube of side (a + b) units. (ii ) The volume of this cube is (a + b)3 cubic units. (iii ) The volume of the two cubes are a 3 and b3 cubic units. (iv ) The volume of each of the cuboid is ab(a + b) cubic units. i.e., the volume of all the three cuboids is 3ab(a + b) cubic units.

Conclusion

⇒ (a + b)3 = a3 + b3 + 3[ab(a + b)] or (a + b)3 = a3 + 3ab(a + b) + b3

Learning Outcomes (i ) The students will obtain the skill of adding the volumes of cubes and cuboids. (ii ) Showing the volume of a cube as the sum of cubes and cuboids helps the students to get a geometric

feeling of volume.

RemarkThe identity (a + b)3 = a3 + b3 + 3ab(a + b) is useful for calculating the cube of a number which can be expressed as the sum of two convenient numbers.

The blocks used in this activity can be cut-out from soft-wood or can be made using cardboard or thermocol sheet.

Note:

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Suggested Activity 1. To verify that (y + z)3 = y3 + z3 + 3y2z + 3yz2 by taking y = 14 and z = 4.

Viva Voce Q1. In the activity of (a + b)3, what do you mean by 3a2b, 3ab2 ? Ans. 3a2b represents sets of cuboids with volumes a × a × b. And, 3ab2 represents three sets of cuboids with volumes a × b × b. Q2. If one side of a cube is (a + 2b), then what is the volume of the cube? Ans. Volume of cube = (a + 2b)3. Q3. What is the degree of (5x + 7y)3? Ans. 3 Q4. What is the coefficient of z2 in (3y +4z)3? Ans. 144y. Q5. Is (a + b)3 binomial? Ans. No. Q6. What is the value of (x + 1)3? Ans. x3 + 3x2 + 3x + 1. Q7. What is an equation? Ans. A statement of equality is called an equation. Q8. What is the difference between an equation and an identity? Ans. An equation is not true for all values of the variable whereas an identity is true for every value of

the variable in it. Q9. The L.H.S. of an identity is (a + b)3. What is its R.H.S.? Ans. a 3 + 3a 2b + 3ab 2 + b 3. Q10. What is the R.H.S. of the identity whose L.H.S. is (a + 2)3? Ans. a 3 + 6a2 + 12a + 8.

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Activity 2.2 Algebraic Identity : (a – b)3 = a3 – 3ab(a – b) – b3

To compute (a – b)3, we extend the identity (a – b)2 = a2 – 2ab + b2 as follows:

(a – b)3 = (a – b)(a – b)2 = (a – b)(a2 – 2ab + b2)

= a(a2 – 2ab + b2) – b(a2 – 2ab + b2) = a3 – 2a2b + ab2 – a2b + 2ab2 – b 3

= a3 – 3a2b + 3ab2 – b3 = a3 – 3ab(a – b) – b3

ObjectiveTo verify the algebraic identity: (a – b)3 = a3 – 3ab(a – b) – b3.

Pre-requisite Knowledge (i ) Volume of a cube = Side × Side × Side (ii ) Volume of a cuboid = Length × Breadth × Height

Procedure (i ) Make a cube of ‘b’ units as shown in figure-1. Its volume is b3 cubic units. Wrap a red glazed paper

on it. (ii ) Make another cube of side (a – b) units as shown in figure 2 and wrap a black glazed paper on it.

Its volume is (a – b)3 cubic units.

Fig. 1 Fig. 2 (iii ) Make three cuboids each of dimensions ‘a’, ‘b’ and (a – b) units as shown in figure 3. Volume of

each cuboid is a × b × (a – b) cubic units. Wrap them with green glazed paper.

Fig. 3

Materials Required Cardboard Glazedpaperofred,greenand

blackcolours Sketchpen,Geometrybox Fevicol,Scale,Cutter,Whitepapers A pair of scissors, Cello tape

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(iv ) Arrange the above five blocks (two cubes and three cuboids) in such a manner that they form a big cube as shown below:

Fig. 4

Observations (i ) Each side of the resultant cube is ‘a’ units. [  (a – b) + b = a] (ii ) Volume of this cube = a3 cubic units. (iii ) This cube is formed by joining the five blocks (2 cubes and 3 cuboids). (iv ) The volume of the two cubes are (a – b)3 cubic units and b3 cubic units. (v ) The volume of the three cuboids taken together is 3ab(a – b) cubic units. (vi ) From the cube of side ‘a’ units, if we remove a cube of volume b3 cubic units (wrapped by a red

glazed paper) and the three cuboids each of volume ab(a – b) cubic units (wrapped by a green glazed paper), then we are left with a cube of side (a – b) units.

Conclusion

⇒ (a – b)3 + b3 + 3[ab(a – b)] = a3

or (a – b)3 = a3 – 3ab(a – b) – b3.

Learning Outcomes (i ) The students will obtain the skill of adding the volume of cubes and cuboids. (ii ) The students will obtain the skill of making a big cube using cuboids and cubes. (iii ) Showing the volume of a cube as the sum of cubes and cuboids helps them to get a geometric feeling

of volume.

RemarkWe call the right hand side expression, i.e., a3 – 3ab(a – b) – b3 the expanded form of the left hand side expression, i.e., (a – b)3 .

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Suggested Activity 1. To verify that (y – z)3 = y3 – 3y2z + 3yz2 – z3 by taking y = 12 and z = 2.

Viva Voce Q1. What is the maximum number of zeros that a cubic polynomial can have? Ans. Three. Q2. Expand (x – 3y)3. Ans. x3 – 27y3 – 9x2y + 27xy2. Q3. For evaluating (999)3, which formula we should use? Ans. We should use (a – b)3 = a3 – b3 – 3ab(a – b) by taking a = 1000 and b = 1. Q4. How would you expand p3 – q3 in terms of (p – q)3? Ans. We know that (p – q)3 = p3 – q3 – 3pq(p – q) = p3 – q3 – 3p2q + 3pq2

⇒ p3 – q3 = (p – q)3 + 3p 2q – 3pq2. Q5. If one side of a cube is given by (a – b), then what is the volume of the cube? Ans. Volume of cube (a – b)3. Q6. Write the coefficient of x3 in (5x – 3y)3. Ans. 125. Q7. The L.H.S. of an identity is (a – b)3. What is its R.H.S.? Ans. a 3 – 3a 2b + 3ab 2 – b 3. Q8. What is the RHS of the identity whose L.H.S. is (a – 1)3? Ans. a 3 – 3a 2 + 3a – 1. Q9. What is the short form of 8x3 + 27y3 + 36x2y + 54xy2? Ans. 8x3 + 27y3 + 36x2y + 54xy2 = (2x)3 + (3y)3 + 3(4x 2)(3y) + 3(2x)(9y 2) = (2x)3 + (3y)3 + 3(2x)2(3y) + 3(2x)(3y)2

= (2x + 3y)3

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