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Mathematics LibreTexts

7.2: Multiplying and Dividing Rational Expressions

  • Anonymous
  • LibreTexts

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Learning Objectives
  • Multiply rational expressions.
  • Divide rational expressions.
  • Multiply and divide rational functions.

Multiplying Rational Expressions

When multiplying fractions, we can multiply the numerators and denominators together and then reduce, as illustrated:

3559=3559=13155193=13

Multiplying rational expressions is performed in a similar manner. For example,

yxxy2=yxxy2=1y1xx1y23=1y

In general, given polynomials P, Q, R, and S, where Q0 and S0, we have

PQRS=PRQS

In this section, assume that all variable expressions in the denominator are nonzero unless otherwise stated.

Example 7.2.1

Multiply:

12x25y320y46x3

Solution:

Multiply numerators and denominators and then cancel common factors.

12x25y320y46x3=240x2y430x3y3Multiply.=82401x2yy4301x3xy31Cancel.=8yx

Answer:

8yx

Example 7.2.2

Multiply:

x3x+5x+5x+7

Solution:

Leave the product in factored form and cancel the common factors.

x3x+5x+5x+7=(x3)(x+5)(x+5)(x+7)=x3x+7

Answer:

x3x+7

Example 7.2.3

Multiply:

15x2y3(2x1)x(2x1)3x2y(x+3)

Solution:

Leave the polynomials in the numerator and denominator factored so that we can cancel the factors. In other words, do not apply the distributive property.

15x2y3(2x1)x(2x1)3x2y(x+3)=15x3y3(2x1)3x2y(2x1)(x+3)Multiply.=515xx3y2y31(2x1)3x2y(2x1)(x+3)Cancel.=5xy2x+3

Answer:

5xy2x+3

Typically, rational expressions will not be given in factored form. In this case, first factor all numerators and denominators completely. Next, multiply and cancel any common factors, if there are any.

Example 7.2.4

Multiply:

x+5x5x5x225

Solution

Factor the denominator x225 as a difference of squares. Then multiply and cancel.

x+5x5x5x225=x+5x5x5(x+5)(x5)Factor.=1(x+5)1(x5)(x5)(x+5)(x5)Cancel.=1x5

Keep in mind that 1 is always a factor; so when the entire numerator cancels out, make sure to write the factor 1.

Answer:

1x5

Example 7.2.5

Multiply:

Solution:

It is a best practice to leave the final answer in factored form.

Answer:

(x+2)(x4)(x2)(x+7)

Example 7.2.6

Multiply:

Solution:

The trinomial 2x2+x+3 in the numerator has a negative leading coefficient. Recall that it is a best practice to first factor out a 1 and then factor the resulting trinomial.

Answer:

3(2x3)x(x+4)

Example 7.2.7

Multiply:

7xx2+3xx2+10x+21x249

Solution:

We replace 7x with 1(x7) so that we can cancel this factor.

7xx2+3xx2+10x+21x249=1(x7)x(x+3)(x+3)(x+7)(x+7)(x7)=1(x7)(x+3)(x+7)x(x+3)(x+7)(x7)=1x=1x

Answer:

1x

Exercise 7.2.1

Multiply:

x2648xx+x2x2+9x+8

Answer

x

Dividing Rational Expressions

To divide two fractions, we multiply by the reciprocal of the divisor, as illustrated:

58÷12=5821=512841=54

Dividing rational expressions is performed in a similar manner. For example,

xy2÷1y=xy2y1=x1yy2y1=xy

In general, given polynomials P, Q, R, and S, where Q0, R0, and S0, we have

PQ÷RS=PQSR=PSQR

Example 7.2.8

Divide:

8x5y25z6÷20xy415z3

Solution:

First, multiply by the reciprocal of the divisor and then cancel.

8x5y25z6÷20xy415z3=8x5y25z615z320xy4Multiplybythereciprocalofthedivisor.=120x5yz3500xy4z6=6120x4x5yz350025xy4y3z6z3Cancel.=6x425y3z3

Answer:

6x425y3z3

Example 7.2.9

Divide:

x+2x24÷x+3x2

Solution:

After multiplying by the reciprocal of the divisor, factor and cancel.

x+2x24÷x+3x2=x+2x24x2x+3Multiplybythereciprocalofthedivisor.=(x+2)(x+2)(x2)(x2)(x+3)Factor.=(x+2)(x2)(x+2)(x2)(x+3)Cancel.=1x+3

Answer:

1x+3

Example 7.2.10

Divide:

Solution:

Begin by multiplying by the reciprocal of the divisor. After doing so, factor and cancel.

Answer:

(x8)(x5)(x+7)2

Example 7.2.11

Divide:

Solution:

Just as we do with fractions, think of the divisor (2x3) as an algebraic fraction over 1.

Answer:

2x+3x+2

Exercise 7.2.2

Divide:

Answer

4x38x2

Multiplying and Dividing Rational Functions

The product and quotient of two rational functions can be simplified using the techniques described in this section. The restrictions to the domain of a product consist of the restrictions of each function.

Example 7.2.12

Calculate (fg)(x) and determine the restrictions to the domain.

Solution:

In this case, the domain of f(x) consists of all real numbers except 0, and the domain of g(x) consists of all real numbers except 14.

Therefore, the domain of the product consists of all real numbers except 0 and 14. Multiply the functions and then simplify the result.

Answer:

(fg)(x)=4x+15x, where x0,14

The restrictions to the domain of a quotient will consist of the restrictions of each function as well as the restrictions on the reciprocal of the divisor.

Example 7.2.13

Calculate (f/g)(x) and determine the restrictions.

Solution:

In this case, the domain of f(x) consists of all real numbers except 3 and 8, and the domain of g(x) consists all real numbers except 3. In addition, the reciprocal of g(x) has a restriction of −8. Therefore, the domain of this quotient consists of all real numbers except 3, 8, and −8.

Answer:

(f/g)(x)=1, where x3,8,8

Key Takeaways

  • After multiplying rational expressions, factor both the numerator and denominator and then cancel common factors. Make note of the restrictions to the domain. The values that give a value of 0 in the denominator are the restrictions.
  • To divide rational expressions, multiply by the reciprocal of the divisor.
  • The restrictions to the domain of a product consist of the restrictions to the domain of each factor.
  • The restrictions to the domain of a quotient consist of the restrictions to the domain of each rational expression as well as the restrictions on the reciprocal of the divisor.
Exercise 7.2.2 Multiplying Rational Expressions

Multiply. (Assume all denominators are nonzero.)

  1. 2x394x2
  2. 5x3yy225x
  3. 5x22y4y215x3
  4. 16a47b249b32a3
  5. x612x324x2x6
  6. x+102x1x2x+10
  7. (y1)2y+11y1
  8. y29y+32y3y3
  9. 2a5a52a+54a225
  10. 2a29a+4a216(a2+4a)
  11. 2x2+3x2(2x1)22xx+2
  12. 9x2+19x+24x2x24x+49x28x1
  13. x2+8x+1616x2x23x4x2+5x+4
  14. x2x2x2+8x+7x2+2x15x25x+6
  15. x+1x33xx+5
  16. 2x1x1x+612x
  17. 9+x3x+13x+9
  18. 12+5x5x+25x
  19. 100y2y1025y2y+10
  20. 3y36y536y2255+6y
  21. 3a2+14a5a2+13a+119a2
  22. 4a216a4a1116a24a215a4
  23. x+9x2+14x45(x281)
  24. 12+5x(25x2+20x+4)
  25. x2+x63x2+15x+182x28x24x+4
  26. 5x24x15x26x+125x210x+1375x2
Answer

1. 32x

3. 2y3x

5. 2x

7. y1y+1

9. 1a5

11. 2x2x1

13. 1

15. x+1x+5

17. 33x+1

19. 25y2

21. a+5a2+1

23. (x+9)2x5

25. 23

Exercise 7.2.3 Dividing Rational Expressions

Divide. (Assume all denominators are nonzero.)

  1. 5x8÷15x24
  2. 38y÷152y2
  3. 5x93y325x109y5
  4. 12x4y221z56x3y27z3
  5. (x4)230x4÷x415x
  6. 5y410(3y5)2÷10y52(3y5)3
  7. x295x÷(x3)
  8. y2648y÷(8+y)
  9. (a8)22a2+10a÷a8a
  10. 24a2b3(a2b)÷12ab(a2b)5
  11. x2+7x+10x2+4x+4÷1x24
  12. 2x2x12x23x+1÷14x21
  13. y+1y23y÷y21y26y+9
  14. 9a2a28a+15÷2a210aa210a+25
  15. a23a182a211a6÷a2+a62a2a1
  16. y27y+10y2+5y14÷2y29y5y2+14y+49
  17. 6y2+y14y2+4y+1÷3y2+2y12y27y4
  18. x27x18x2+8x+12÷x281x2+12x+36
  19. 4a2b2b+2a÷(b2a)2
  20. x2y2y+x÷(yx)2
  21. 5y2(y3)4x3÷25y(3y)2x2
  22. 15x33(y+7)÷25x69(7+y)2
  23. 3x+4x8÷7x8x
  24. 3x22x+1÷23x3x
  25. (7x1)24x+1÷28x211x+114x
  26. 4x(x+2)2÷2xx24
  27. a2b2a÷(ba)2
  28. (a2b)22b÷(2b2+aba2)
  29. x26x+9x2+7x+12÷9x2x2+8x+16
  30. 2x29x525x2÷14x+4x22x29x+5
  31. 3x216x+51004x2÷9x26x+13x2+14x5
  32. 10x225x15x26x+9÷9x2x2+6x+9
Answer

1. 16x

3. 3y25x

5. x42x3

7. x+35x

9. a82(a+5)

11. (x+5)(x2)

13. y3y(y1)

15. a1a2

17. y4y+1

19. 12ab

21. y10x

23. 3x+47x

25. 7x14x+1

27. a+ba(ab)

29. (x3)(x+4)(x+3)2

31. 14

Exercise 7.2.4 Dividing Rational Expressions

Recall that multiplication and division are to be performed in the order they appear from left to right. Simplify the following.

  1. 1x2x1x+3÷x1x3
  2. x7x+91x3÷x7x
  3. x+1x2÷xx5x2x+1
  4. x+42x+5÷x32x+5x+4x3
  5. 2x1x+1÷x4x2+1x42x1
  6. 4x213x+2÷2x1x+53x+22x+1
Answer

1. xx+3

3. x(x5)x2

5. x2+1x+1

Exercise 7.2.5 Multiplying and Dividing Rational Functions

Calculate (fg)(x) and determine the restrictions to the domain.

  1. f(x)=1x and g(x)=1x1
  2. f(x)=x+1x1 and g(x)=x21
  3. f(x)=3x+2x+2 and g(x)=x24(3x+2)2
  4. f(x)=(13x)2x6 and g(x)=(x6)29x21
  5. f(x)=25x21x2+6x+9 and g(x)=x295x+1
  6. f(x)=x2492x2+13x7 and g(x)=4x24x+17x
Answer

1. (fg)(x)=1x(x1);x0,1

3. (fg)(x)=x23x+2;x2,23

5. (fg)(x)=(x3)(5x1)x+3;x3,15

Exercise 7.2.6 Multiplying and Dividing Rational Functions

Calculate (f/g)(x) and state the restrictions.

  1. f(x)=1x and g(x)=x2x1
  2. f(x)=(5x+3)2x2 and g(x)=5x+36x
  3. f(x)=5x(x8)2 and g(x)=x225x8
  4. f(x)=x22x15x23x10 and g(x)=2x25x3x27x+12
  5. f(x)=3x2+11x49x26x+1 and g(x)=x22x+13x24x+1
  6. f(x)=36x2x2+12x+36 and g(x)=x212x+36x2+4x12
Answer

1. (f/g)(x)=x1x(x2);x0,1,2

3. (f/g)(x)=1(x8)(x+5);x±5,8

5. (f/g)(x)=(x+4)(x1);x13,1

Exercise 7.2.7 Discussion Board Topics
  1. In the history of fractions, who is credited for the first use of the fraction bar?
  2. How did the ancient Egyptians use fractions?
  3. Explain why x=7 is a restriction to 1x÷x7x2.
Answer

1. Answer may vary

3. Answer may vary


This page titled 7.2: Multiplying and Dividing Rational Expressions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform.

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