Multiply and Divide Rational Expressions* - OpenStax CNX

OpenStax-CNX module: m63367 1

Multiply and Divide Rational

Expressions*

OpenStax

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Creative Commons Attribution License 4.0�

Abstract

By the end of this section, you will be able to:

• Determine the values for which a rational expression is unde�ned

• Simplify rational expressions

• Multiply rational expressions

• Divide rational expressions

• Multiply and divide rational functions

note: Before you get started, take this readiness quiz.

1.Simplify: 90y15y2 .

If you missed this problem, review .2.Multiply: 14

15 ·635 .

If you missed this problem, review .3.Divide: 12

10 ÷825 .

If you missed this problem, review .

We previously reviewed the properties of fractions and their operations. We introduced rational numbers,which are just fractions where the numerators and denominators are integers. In this chapter, we willwork with fractions whose numerators and denominators are polynomials. We call this kind of expression arational expression.

note: A rational expression is an expression of the form pq , where p and q are polynomials and

q 6= 0.

Here are some examples of rational expressions:

− 2456

5x12y

4x+1x2−9

4x2+3x−12x−8

(1)

Notice that the �rst rational expression listed above, − 2456 , is just a fraction. Since a constant is a polynomial

with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.We will do the same operations with rational expressions that we did with fractions. We will simplify,

add, subtract, multiply, divide and use them in applications.

*Version 1.4: Mar 10, 2017 10:03 am -0600�http://creativecommons.org/licenses/by/4.0/

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1 Determine the Values for Which a Rational Expression is Unde�ned

If the denominator is zero, the rational expression is unde�ned. The numerator of a rational expression maybe 0�but not the denominator.

When we work with a numerical fraction, it is easy to avoid dividing by zero because we can see thenumber in the denominator. In order to avoid dividing by zero in a rational expression, we must not allowvalues of the variable that will make the denominator be zero.

So before we begin any operation with a rational expression, we examine it �rst to �nd the values thatwould make the denominator zero. That way, when we solve a rational equation for example, we will knowwhether the algebraic solutions we �nd are allowed or not.

note:

Step 1.Set the denominator equal to zero.Step 2.Solve the equation.

Example 1Determine the value for which each rational expression is unde�ned:

a 8a2b3c b 4b−3

2b+5 c x+4x2+5x+6 .

SolutionThe expression will be unde�ned when the denominator is zero.

a8a2b3c

Set the denominator equal to zero and solve

for the variable.3c = 0

c = 0

8a2b3c is unde�ned for c = 0.

b4b−32b+5


for the variable. 2b+ 5 = 0

2b = −5b = − 5

2

4b−32b+5 is unde�ned for b = − 5

2 .c

x+4x2+5x+6


for the variable.

x2 + 5x+ 6 = 0

(x+ 2) (x+ 3) = 0

x+ 2 = 0 or x+ 3 = 0

x = −2 or x = −3x+4

x2+5x+6 is unde�ned for x = −2 or x = −3.



note:

Exercise 2 (Solution on p. 21.)

Determine the value for which each rational expression is unde�ned.

a 3y2

8x b 8n−53n+1 c a+10

a2+4a+3

note:


Determine the value for which each rational expression is unde�ned.

a 4p5q b y−1

3y+2 c m−5m2+m−6

2 Simplify Rational Expressions

A fraction is considered simpli�ed if there are no common factors, other than 1, in its numerator anddenominator. Similarly, a simpli�ed rational expression has no common factors, other than 1, in itsnumerator and denominator.

note: A rational expression is considered simpli�ed if there are no common factors in its numeratorand denominator.

For example,

x+2x+3 is simpli�ed because there are no common factors of x+ 2 and x+ 3.

2x3x is not simpli�ed because x is a common factor of 2x and 3x.

(2)

We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will alsouse it to simplify rational expressions.

note: If a, b, and c are numbers where b 6= 0, c 6= 0,

thena

b=

a · cb · c

anda · cb · c

=a

b. (3)

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero arespeci�cally disallowed. We see b 6= 0, c 6= 0 clearly stated.

To simplify rational expressions, we �rst write the numerator and denominator in factored form. Thenwe remove the common factors using the Equivalent Fractions Property.

Be very careful as you remove common factors. Factors are multiplied to make a product. You canremove a factor from a product. You cannot remove a term from a sum.



Remov-ing the x's from x+5

x would be like cancelling the 2's in the fraction 2+52 !

Example 2: How to Simplify a Rational Expression

Simplify: x2+5x+6x2+8x+12 .

Solution

note: Exercise 5 (Solution on p. 21.)

Simplify: x2−x−2x2−3x+2 .

note:


Simplify: x2−3x−10x2+x−2 .

We now summarize the steps you should follow to simplify rational expressions.

note:



Step 1.Factor the numerator and denominator completely.Step 2.Simplify by dividing out common factors.

Usually, we leave the simpli�ed rational expression in factored form. This way, it is easy to check that wehave removed all the common factors.

We'll use the methods we have learned to factor the polynomials in the numerators and denominators inthe following examples.

Every time we write a rational expression, we should make a statement disallowing values that wouldmake a denominator zero. However, to let us focus on the work at hand, we will omit writing it in theexamples.

Example 3

Simplify: 3a2−12ab+12b2

6a2−24b2 .

Solution3a2−12ab+12b2

6a2−24b2

Factor the numerator and denominator,

�rst factoring out the GCF.

3(a2−4ab+4b2)6(a2−4b2)

3(a−2b)(a−2b)6(a+2b)(a−2b)

Remove the common factors of a− 2b and 3. )3(a−2b))(a−2b)

)3·2(a+2b))(a−2b)

a−2b2(a+2b)

note:


Simplify: 2x2−12xy+18y2

3x2−27y2 .


Simplify: 5x2−30xy+25y2

2x2−50y2 .

Now we will see how to simplify a rational expression whose numerator and denominator have oppositefactors. We previously introduced opposite notation: the opposite of a is −a and −a = −1 · a.

The numerical fraction, say 7−7 simpli�es to −1. We also recognize that the numerator and denominator

are opposites.The fraction a

−a , whose numerator and denominator are opposites also simpli�es to −1.



Let's look at the expression b− a. b− a

Rewrite. −a+ b

Factor out −−1. − 1 (a− b)

(4)

This tells us that b− a is the opposite of a− b.In general, we could write the opposite of a− b as b− a. So the rational expression a−b

b−a simpli�es to −1.

note: The opposite of a− b is b− a.

a− b

b− a= −1 a 6= b (5)

An expression and its opposite divide to −1.

We will use this property to simplify rational expressions that contain opposites in their numerators anddenominators. Be careful not to treat a + b and b + a as opposites. Recall that in addition, order doesn'tmatter so a+ b = b+ a. So if a 6= −b, then a+b

b+a = 1.

Example 4

Simplify: x2−4x−3264−x2 .

Solution

Factor the numerator and the denominator.

Recognize the factors that are opposites.

Simplify.

Table 1


Simplify: x2−4x−525−x2 .

note:


Simplify: x2+x−21−x2 .



3 Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numer-ators and multiply the denominators. Then, if there are any common factors, we remove them to simplifythe result.

note: If p, q, r, and s are polynomials where q 6= 0, s 6= 0, then

p

q· rs=

pr

qs(6)

To multiply rational expressions, multiply the numerators and multiply the denominators.

Remember, throughout this chapter, we will assume that all numerical values that would make the denomi-nator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mindthat the denominator can never be zero. So in this next example, x 6= 0,x 6= 3, and x 6= 4.

Example 5: How to Multiply Rational Expressions

Simplify: 2xx2−7x+12 ·

x2−96x2 .

Solution


Simplify: 5xx2+5x+6 ·

x2−410x .

note:


Simplify: 9x2

x2+11x+30 ·x2−363x2 .



note:

Step 1.Factor each numerator and denominator completely.Step 2.Multiply the numerators and denominators.Step 3.Simplify by dividing out common factors.

Example 6

Multiply: 3a2−8a−3a2−25 · a

2+10a+253a2−14a−5 .

Solution3a2−8a−3

a2−25 · a2+10a+25

3a2−14a−5

Factor the numerators and denominators

and then multiply.

(3a+1)(a−3)(a+5)(a+5)(a−5)(a+5)(3a+1)(a−5)

Simplify by dividing out

common factors.

)(3a+1)(a−3))(a+5)(a+5)

(a−5))(a+5))(3a+1)(a−5)

Simplify. (a−3)(a+5)(a−5)(a−5)

Rewrite (a− 5) (a− 5) using an exponent. (a−3)(a+5)

(a−5)2

note:


Simplify: 2x2+5x−12x2−16 · x2−8x+16

2x2−13x+15 .

note:


Simplify: 4b2+7b−21−b2 · b2−2b+1

4b2+15b−4 .

4 Divide Rational Expressions

Just like we did for numerical fractions, to divide rational expressions, we multiply the �rst fraction by thereciprocal of the second.



note: If p, q, r, and s are polynomials where q 6= 0, r 6= 0, s 6= 0, then

p

q÷ r

s=

p

q· sr

(7)

To divide rational expressions, multiply the �rst fraction by the reciprocal of the second.

Once we rewrite the division as multiplication of the �rst expression by the reciprocal of the second, we thenfactor everything and look for common factors.

Example 7: How to Divide Rational Expressions

Divide: p3+q3

2p2+2pq+2q2 ÷p2−q2

6 .Solution


Simplify: x3−83x2−6x+12 ÷

x2−46 .

note:


Simplify: 2z2

z2−1 ÷z3−z2+z

z3+1 .

note:



Step 1.Rewrite the division as the product of the �rst rational expression and the reciprocal of thesecond.

Step 2.Factor the numerators and denominators completely.Step 3.Multiply the numerators and denominators together.Step 4.Simplify by dividing out common factors.

Recall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in thenumerator, the denominator or both. Also, remember a fraction bar means division. A complex fraction isanother way of writing division of two fractions.

Example 8

Divide:6x2−7x+2

4x−8

2x2−7x+3

x2−5x+6

.

Solution6x2−7x+2

4x−8

2x2−7x+3

x2−5x+6

Rewrite with a division sign. 6x2−7x+24x−8 ÷ 2x2−7x+3

x2−5x+6

Rewrite as product of �rst times reciprocal

of second.

6x2−7x+24x−8 · x2−5x+6

2x2−7x+3

Factor the numerators and the

denominators, and then multiply.

(2x−1)(3x−2)(x−2)(x−3)4(x−2)(2x−1)(x−3)

Simplify by dividing out common factors. )(2x−1)(3x−2))(x−2))(x−3)

4)(x−2))(2x−1))(x−3)

Simplify. 3x−24

note:


Simplify:3x2+7x+2

4x+24

3x2−14x−5

x2+x−30

.



note:


Simplify:y2−36

2y2+11y−6

2y2−2y−608y−4

.

If we have more than two rational expressions to work with, we still follow the same procedure. The �rststep will be to rewrite any division as multiplication by the reciprocal. Then, we factor and multiply.

Example 9

Perform the indicated operations: 3x−64x−4 ·

x2+2x−3x2−3x−10 ÷

2x+128x+16 .

Solution

Rewrite the division as multiplicationby the reciprocal.

Factor the numerators and the denominators.

Multiply the fractions. Bringing the constants tothe front will help when removing common factors.

Simplify by dividing out common factors.

Simplify.

Table 2


Perform the indicated operations: 4m+43m−15 ·

m2−3m−10m2−4m−32 ÷

12m−366m−48 .



note:


Perform the indicated operations: 2n2+10nn−1 ÷ n2+10n+24

n2+8n−9 ·n+4

8n2+12n .

5 Multiply and Divide Rational Functions

We started this section stating that a rational expression is an expression of the form pq , where p and q

are polynomials and q 6= 0. Similarly, we de�ne a rational function as a function of the form R (x) = p(x)q(x)

where p (x) and q (x) are polynomial functions and q (x) is not zero.

note: A rational function is a function of the form

R (x) =p (x)

q (x)(8)

where p (x) and q (x) are polynomial functions and q (x) is not zero.

The domain of a rational function is all real numbers except for those values that would cause division byzero. We must eliminate any values that make q (x) = 0.

note:

Step 1.Set the denominator equal to zero.Step 2.Solve the equation.Step 3.The domain is all real numbers excluding the values found in Step 2.

Example 10

Find the domain of R (x) = 2x2−14x4x2−16x−48 .

SolutionThe domain will be all real numbers except those values that make the denominator zero. We willset the denominator equal to zero , solve that equation, and then exclude those values from thedomain.

Set the denominator to zero. 4x2 − 16x− 48 = 0

Factor, �rst factor out the GCF. 4(x2 − 4x− 12

)= 0

4 (x− 6) (x+ 2) = 0

Use the Zero Product Property. 4 6= 0 x− 6 = 0 x+ 2 = 0

Solve. x = 6 x = −2The domain ofR (x) is all real numbers

where x 6= 6 and x 6= −2.

note:





note:



To multiply rational functions, we multiply the resulting rational expressions on the right side of the equationusing the same techniques we used to multiply rational expressions.

Example 11

Find R (x) = f (x) · g (x) where f (x) = 2x−6x2−8x+15 and g (x) = x2−25

2x+10 .

Solution

R (x) = f (x) · g (x)

R (x) = 2x−6x2−8x+15 ·

x2−252x+10

Factor each numerator and denominator. R (x) = 2(x−3)(x−3)(x−5) ·

(x−5)(x+5)2(x+5)

Multiply the numerators and denominators. R (x) = 2(x−3)(x−5)(x+5)2(x−3)(x−5)(x+5)

Remove common factors. R (x) = )2)(x−3))(x−5))(x+5)

)2)(x−3))(x−5))(x+5)

Simplify. R (x) = 1

note:


Find R (x) = f (x) · g (x) where f (x) = 3x−21x2−9x+14 and g (x) = 2x2−8

3x+6 .

note:


Find R (x) = f (x) · g (x) where f (x) = x2−x3x2+27x−30 and g (x) = x2−100

x2−10x .

To divide rational functions, we divide the resulting rational expressions on the right side of the equationusing the same techniques we used to divide rational expressions.



Example 12

Find R (x) = f(x)g(x) where f (x) = 3x2

x2−4x and g (x) = 9x2−45xx2−7x+10 .

Solution

R (x) = f(x)g(x)

Substitute in the functions f (x) , g (x) . R (x) =3x2

x2−4x

9x2−45x

x2−7x+10

Rewrite the division as the product of

f (x) and the reciprocal of g (x) .R (x) = 3x2

x2−4x ·x2−7x+109x2−45x

Factor the numerators and denominators

and then multiply.R (x) = 3·x·x·(x−5)(x−2)

x(x−4)·3·3·x·(x−5)

Simplify by dividing out common factors. R (x) = )3·)x·)x)(x−5)(x−2)

)x(x−4)·)3·3·)x)(x−5)

R (x) = x−23(x−4)

note:



x2−8x and g (x) = 8x2+24xx2+x−6 .

note:



3x2+33x and g (x) = 5x−5x2+9x−22 .

6 Key Concepts

� Determine the values for which a rational expression is unde�ned.

Step a. Set the denominator equal to zero.Step b. Solve the equation.

� Equivalent Fractions PropertyIf a, b, and c are numbers where b 6= 0, c 6= 0, then a

b = a·cb·c and a·c

b·c = ab .

� How to simplify a rational expression.



Step a. Factor the numerator and denominator completely.Step b. Simplify by dividing out common factors.

� Opposites in a Rational ExpressionThe opposite of a− b is b− a.a−bb−a = −1 a 6= bAn expression and its opposite divide to −1.

� Multiplication of Rational ExpressionsIf p, q, r, and s are polynomials where q 6= 0, s 6= 0, then

pq ·

rs = pr

qs� How to multiply rational expressions.

Step a. Factor each numerator and denominator completely.Step b. Multiply the numerators and denominators.Step c. Simplify by dividing out common factors.

� Division of Rational ExpressionsIf p, q, r, and s are polynomials where q 6= 0, r 6= 0, s 6= 0, then

pq ÷

rs = p

q ·sr

� How to divide rational expressions.

Step a. Rewrite the division as the product of the �rst rational expression and the reciprocal of the second.Step b. Factor the numerators and denominators completely.Step c. Multiply the numerators and denominators together.Step d. Simplify by dividing out common factors.

� How to determine the domain of a rational function.

Step a. Set the denominator equal to zero.Step b. Solve the equation.Step c. The domain is all real numbers excluding the values found in Step 2.

7

7.1 Practice Makes Perfect

Determine the Values for Which a Rational Expression is Unde�nedIn the following exercises, determine the values for which the rational expression is unde�ned.


a 2x2

z

b 4p−16p−5

c n−3n2+2n−8

Exercise 38

a 10m11n

b 6y+134y−9

c b−8b2−36


a 4x2y3y

b 3x−22x+1

c u−1u2−3u−28



Exercise 40

a 5pq2

9q

b 7a−43a+5

c 1x2−4

Simplify Rational ExpressionsIn the following exercises, simplify each rational expression.


− 4455

Exercise 425663

Exercise 43 (Solution on p. 22.)8m3n12mn2

Exercise 4436v3w2

27vw3

Exercise 45 (Solution on p. 22.)8n−963n−36

Exercise 4612p−2405p−100

Exercise 47 (Solution on p. 22.)x2+4x−5x2−2x+1

Exercise 48y2+3y−4y2−6y+5

Exercise 49 (Solution on p. 22.)a2−4

a2+6a−16

Exercise 50y2−2y−3

y2−9

Exercise 51 (Solution on p. 22.)p3+3p2+4p+12

p2+p−6

Exercise 52x3−2x2−25x+50

x2−25

Exercise 53 (Solution on p. 22.)8b2−32b

2b2−6b−80

Exercise 54−5c2−10c

−10c2+30c+100

Exercise 55 (Solution on p. 22.)3m2+30mn+75n2

4m2−100n2

Exercise 565r2+30rs−35s2

r2−49s2

Exercise 57 (Solution on p. 22.)a−55−a

Exercise 585−dd−5

Exercise 59 (Solution on p. 22.)20−5yy2−16



Exercise 604v−3264−v2

Exercise 61 (Solution on p. 22.)w3+216w2−36

Exercise 62v3+125v2−25

Exercise 63 (Solution on p. 22.)z2−9z+20

16−z2

Exercise 64a2−5z−36

81−a2

Multiply Rational ExpressionsIn the following exercises, multiply the rational expressions.

Exercise 65 (Solution on p. 22.)1216 ·

410

Exercise 66325 ·

1624

Exercise 67 (Solution on p. 22.)5x2y4

12xy3 · 6x2

20y2

Exercise 6812a3bb2 · 2ab

2

9b3

Exercise 69 (Solution on p. 22.)5p2

p2−5p−36 ·p2−1610p

Exercise 703q2

q2+q−6 ·q2−99q

Exercise 71 (Solution on p. 22.)2y2−10y

y2+10y+25 ·y+56y

Exercise 72z2+3z

z2−3z−4 ·z−4z2

Exercise 73 (Solution on p. 22.)28−4b3b−3 ·

b2+8b−9b2−49

Exercise 7472m−12m2

8m+32 · m2+10m+24m2−36

Exercise 75 (Solution on p. 22.)5c2+9c+2c2−25 · c

2+10c+253c2−14c−5

Exercise 762d2+d−3d2−16 ·

d2−8d+162d2−9d−18

Exercise 77 (Solution on p. 22.)6m2−2m−10

9−m2 · m2−6m+96m2+29m−20

Exercise 782n2−3n−14

25−n2 · n2−10n+252n2−13n+21

Divide Rational ExpressionsIn the following exercises, divide the rational expressions.

Exercise 79 (Solution on p. 22.)v−511−v ÷

v2−25v−11



Exercise 8010+ww−8 ÷

100−w2

8−w

Exercise 81 (Solution on p. 23.)3s2

s2−16 ÷s3−4s2+16s

s3−64

Exercise 82r2−915 ÷

r3−275r2+15r+45

Exercise 83 (Solution on p. 23.)p3+q3

3p2+3pq+3q2 ÷p2−q2

12

Exercise 84v3−8w3

2v2+4vw+8w2 ÷ v2−4w2

4

Exercise 85 (Solution on p. 23.)x2+3x−10

4x ÷(2x2 + 20x+ 50

)Exercise 86

2y2−10yz−48z2

2y−1 ÷(4y2 − 32yz

)Exercise 87 (Solution on p. 23.)

2a2−a−215a+20

a2+7a+12

a2+8a+16

Exercise 883b2+2b−812b+18

3b2+2b−8

2b2−7b−15

Exercise 89 (Solution on p. 23.)12c2−12

2c2−3c+14c+4

6c2−13c+5

Exercise 904d2+7d−235d+10

d2−4

7d2−12d−4

For the following exercises, perform the indicated operations.

Exercise 91 (Solution on p. 23.)10m2+80m

3m−9 · m2+4m−21

m2−9m+20 ÷5m2+10m2m−10

Exercise 924n2+32n3n+2 · 3n

2−n−2n2+n−30 ÷

108n2−24nn+6

Exercise 93 (Solution on p. 23.)12p2+3p

p+3 ÷ p2+2p−63p2−p−12 ·

p−79p3−9p2

Exercise 946q+3

9q2−9q ÷q2+14q+33q2+4q−5 ·

4q2+12q12q+6

Multiply and Divide Rational FunctionsIn the following exercises, �nd the domain of each function.


R (x) = x3−2x2−25x+50x2−25

Exercise 96R (x) = x3+3x2−4x−12

x2−4


R (x) = 3x2+15x6x2+6x−36

Exercise 98R (x) = 8x2−32x

2x2−6x−80



For the following exercises, �nd R (x) = f (x) · g (x) where f (x) and g (x) are given.


f (x) = 6x2−12xx2+7x−18

g (x) = x2−813x2−27x

Exercise 100f (x) = x2−2x

x2+6x−16

g (x) = x2−64x2−8x


f (x) = 4xx2−3x−10

g (x) = x2−258x2

Exercise 102f (x) = 2x2+8x

x2−9x+20

g (x) = x−5x2

For the following exercises, �nd R (x) = f(x)g(x) where f (x) and g (x) are given.


f (x) = 27x2

3x−21

g (x) = 3x2+18xx2+13x+42

Exercise 104f (x) = 24x2

2x−8

g (x) = 4x3+28x2

x2+11x+28


f (x) = 16x2

4x+36

g (x) = 4x2−24xx2+4x−45

Exercise 106f (x) = 24x2

2x−4

g (x) = 12x2+36xx2−11x+18

7.2 Writing Exercises


Explain how you �nd the values of x for which the rational expression x2−x−20x2−4 is unde�ned.

Exercise 108Explain all the steps you take to simplify the rational expression p2+4p−21

9−p2 .


a Multiply 74 ·

910 and explain all your steps.

b Multiply nn−3 ·

9n+3 and explain all your steps.

c Evaluate your answer to part b when n = 7. Did you get the same answer you got in part a?Why or why not?

Exercise 110a Divide 24

5 ÷ 6 and explain all your steps.

b Divide x2−1x ÷ (x+ 1) and explain all your steps.

c Evaluate your answer to part b when x = 5. Did you get the same answer you got in part a? Whyor why not?



7.3 Self Check

a After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b Ifmost of your checks were:

. . .con�dently. Congratulations! You have achieved your goals in this section! Re�ect on the studyskills you used so that you can continue to use them. What did you do to become con�dent of your abilityto do these things? Be speci�c!

. . .with some help. This must be addressed quickly as topics you do not master become potholes inyour road to success. Math is sequential - every topic builds upon previous work. It is important to makesure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmatesand instructor are good resources. Is there a place on campus where math tutors are available? Can yourstudy skills be improved?

. . .no - I don't get it! This is critical and you must not ignore it. You need to get help immediately oryou will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Togetheryou can come up with a plan to get you the help you need.



Solutions to Exercises in this Module

Solution to Exercise (p. 3)a x = 0 b n = − 1

3c a = −1, a = −3Solution to Exercise (p. 3)a q = 0 b y = − 2

3c m = 2,m = −3Solution to Exercise (p. 4)x+1x−1 , x 6= 2, x 6= 1Solution to Exercise (p. 4)x−5x−1 , x 6= −2, x 6= 1Solution to Exercise (p. 5)2(x−3y)3(x+3y)

Solution to Exercise (p. 5)5(x−y)2(x+5y)

Solution to Exercise (p. 6)−x+1

x+5Solution to Exercise (p. 6)−x+2

x+1Solution to Exercise (p. 7)

x−22(x+3)

Solution to Exercise (p. 7)3(x−6)x+5

Solution to Exercise (p. 8)x−4x−5Solution to Exercise (p. 8)

− (b+2)(b−1)(1+b)(b+4)

Solution to Exercise (p. 9)2(x2+2x+4)

(x+2)(x2−2x+4)

Solution to Exercise (p. 9)2zz−1Solution to Exercise (p. 10)x+24

Solution to Exercise (p. 11)2

y+5

Solution to Exercise (p. 11)2(m+1)(m+2)3(m+4)(m−3)

Solution to Exercise (p. 12)(n+5)(n+9)

2(n+6)(2n+3)

Solution to Exercise (p. 12)The domain of R (x) is all real numbers where x 6= 5 and x 6= −1.Solution to Exercise (p. 13)The domain of R (x) is all real numbers where x 6= 4 and x 6= −2.Solution to Exercise (p. 13)R (x) = 2Solution to Exercise (p. 13)R (x) = 1

3



Solution to Exercise (p. 14)R (x) = x−2

4(x−8)

Solution to Exercise (p. 14)

R (x) = x(x−2)x−1

Solution to Exercise (p. 15)a z = 0 b p = 5

6c n = −4, n = 2Solution to Exercise (p. 15)a y = 0 b x = − 1

2c u = −4, u = 7Solution to Exercise (p. 16)− 4

5Solution to Exercise (p. 16)2m2

3nSolution to Exercise (p. 16)83Solution to Exercise (p. 16)x+5x−1Solution to Exercise (p. 16)a+2a+8Solution to Exercise (p. 16)p2+4p−2

Solution to Exercise (p. 16)4b(b−4)

(b+5)(b−8)

Solution to Exercise (p. 16)3(m+5n)4(m−5n)

Solution to Exercise (p. 16)−1Solution to Exercise (p. 16)− 5

y+4

Solution to Exercise (p. 17)w2−6w+36

w−6Solution to Exercise (p. 17)− z−5

4+zSolution to Exercise (p. 17)310Solution to Exercise (p. 17)x3

8y

Solution to Exercise (p. 17)p(p−4)2(p−9)

Solution to Exercise (p. 17)y−5

3(y+5)


− 4(b+9)3(b+7)

Solution to Exercise (p. 17)(c+2)(c+2)(c−2)(c−3)


− (m−2)(m−3)(3+m)(m+4)



Solution to Exercise (p. 17)− 1

v+5Solution to Exercise (p. 18)

3ss+4Solution to Exercise (p. 18)

4(p2−pq+q2)(p−q)(p2+pq+q2)

Solution to Exercise (p. 18)x−28x

Solution to Exercise (p. 18)2a−7

5Solution to Exercise (p. 18)3 (3c− 5)Solution to Exercise (p. 18)4(m+8)(m+7)3(m−4)(m+2)

Solution to Exercise (p. 18)(4p+1)(p−7)3p(p+9)(p−1)

Solution to Exercise (p. 18)x 6= 5 and x 6= −5Solution to Exercise (p. 18)x 6= 2 and x 6= −3Solution to Exercise (p. 19)R (x) = 2Solution to Exercise (p. 19)R (x) = x+5

2x(x+2)


R (x) = 3x(x+7)x−7


R (x) = x(x−5)x−6

Solution to Exercise (p. 19)Answers will vary.Solution to Exercise (p. 19)Answers will vary.

Glossary

De�nition 8: rational expressionA rational expression is an expression of the form p

q , where p and q are polynomials and q 6= 0.

De�nition 8: simpli�ed rational expressionA simpli�ed rational expression has no common factors, other than 1, in its numerator and denom-inator.

De�nition 8: rational functionA rational function is a function of the form R (x) = p(x)

q(x) where p (x) and q (x) are polynomial

functions and q (x) is not zero.


Multiply and Divide Rational Expressions* - OpenStax CNX

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