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12.5: Multiplication and Division of Rational Expressions

Difficulty Level: At Grade Created by: CK-12

Because a rational expression is really a fraction, two (or more) rational expressions can be combined through multiplication and/or division in the same manner as numerical fractions. A reminder of how to multiply fractions is below.

For any rational expressions a0,b0,c0,d0,

abcd=acbdab÷cdabdc=adbc

Example: Multiply the following a16b84b35a2

Solution:

a16b84b35a24ab380a2b8

Simplify exponents using methods learned in chapter 8.

4ab380a2b8=120ab5

Example 1: Simplify 9c24y221c4.

Solution:

9c24y221c49c214y221c49c214y221c4=36c2y221c436c2y221c4=12y27c2

Multiplying Rational Expressions Involving Polynomials

When rational expressions become complex, it is usually easier to factor and reduce them before attempting to multiply the expressions.

Example: Multiply 4x+123x2xx29.

Solution: Factor all pieces of these rational expressions and reduce before multiplying.

4x+123x2xx294(x+3)3x2x(x+3)(x3)4(x+3)3x2x(x+3)(x3)43x1x343x29x

Example 1: Multiply 12x2x6x21x2+7x+64x227x+18.

Solution: Factor all pieces, reduce, and then multiply.

12x2x6x21x2+7x+64x227x+18(3x+2)(4x3)(x+1)(x1)(x+1)(x+6)(4x3)(x6)12x2x6x21x2+7x+64x227x+18(3x+2)(4x3)(x+1)(x1)(x+1)(x+6)(4x3)(x6)(3x+2)(x+6)(x1)(x6)=3x2+20x+12x27x+6

Dividing Rational Expressions Involving Polynomials

Division of rational expressions works in the same manner as multiplication. A reminder of how to divide fractions is below.

For any rational expressions a0,b0,c0,d0,

ab÷cdabdc=adbc

Example: Simplify 9x242x2÷21x22x81.

Solution:

9x242x2÷21x22x819x242x2121x22x8

Repeat the process for multiplying rational expressions.

9x242x2121x22x89x242x2÷21x22x81(3x2)(3x2)2(x1)1(3x2)(7x+4)=3x214x26x8

Real-Life Application

Suppose Marciel is training for a running race. Marciel’s speed (in miles per hour) of his training run each morning is given by the function x39x, where x is the number of bowls of cereal he had for breakfast (1x6). Marciel’s training distance (in miles), if he eats x bowls of cereal, is 3x29x. What is the function for Marciel’s time and how long does it take Marciel to do his training run if he eats five bowls of cereal on Tuesday morning?

timetimetimeIf xtime=distancespeed=3x29xx39x=3x(x3)x(x29)=3x(x3)x(x+3)(x3)=3x+3=5,then=35+3=38
.

Marciel will run for 38 of an hour.

Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both.

CK-12 Basic Algebra: Multiplying and Dividing Rational Expressions (9:19)

In 1–20, perform the indicated operation and reduce the answer to lowest terms

  1. x32y32y2x
  2. 2xy÷2x2y
  3. 2xy24y5x
  4. 2xy2y2x3
  5. 4y21y29y32y1
  6. 6aba2a3b3b2
  7. x2x1÷xx2+x2
  8. 33a252011a3
  9. a2+2ab+b2ab2a2b÷(a+b)
  10. 2x2+2x24x2+3xx2+x6x+4
  11. 3x3x5÷x292x28x10
  12. x225x+3÷(x5)
  13. 2x+12x1÷4x2112x
  14. xx5x28x+15x23x
  15. 3x2+5x12x29÷3x43x+4
  16. 5x2+16x+336x225(6x2+5x)
  17. x2+7x+10x29x23x3x2+4x4
  18. x2+x12x2+4x+4÷x3x+2
  19. x416x29÷x2+4x2+6x+9
  20. x2+8x+167x2+9x+2÷7x+2x2+4x
  21. Maria’s recipe asks for 212 times more flour than sugar. How many cups of flour should she mix in if she uses 313 cups of sugar?
  22. George drives from San Diego to Los Angeles. On the return trip, he increases his driving speed by 15 miles per hour. In terms of his initial speed, by what factor is the driving time decreased on the return trip?
  23. Ohm’s Law states that in an electrical circuit I=VRc. The total resistance for resistors placed in parallel is given by 1Rtot=1R1+1R2. Write the formula for the electric current in term of the component resistances: R1 and R2.

Mixed Review

  1. The time it takes to reach a destination varies inversely as the speed in which you travel. It takes 3.6 hours to reach your destination traveling 65 miles per hour. How long would it take to reach your destination traveling 78 miles per hour?
  2. A local nursery makes two types of fall arrangements. One arrangement uses eight mums and five black-eyed susans. The other arrangement uses six mums and 9 black-eyed susans. The nursery can use no more than 144 mums and 135 black-eyed susans. The first arrangement sells for $49.99 and the second arrangement sells for 38.95. How many of each type should be sold to maximize revenue?
  3. Solve for r and graph the solution on a number line: 24|2r+3|.
  4. What is true of any line parallel to 5x+9y=36?
  5. Solve for d:3+5d=d(3x3).
  6. Graph and determine the domain and range: y9=x25x.
  7. Rewrite in vertex form by completing the square. Identify the vertex: y216y+3=4.

Quick Quiz

  1. h is inversely proportional to t. If t=0.05153 when h=16, find t when h=1.45.
  2. Use f(x)=5x225 for the following questions.
    1. Find the excluded values.
    2. Determine the vertical asymptotes.
    3. Sketch a graph of this function.
    4. Determine its domain and range.
  3. Simplify 8c4+12c222c+14.
  4. Simplify 10a230aa3. What are its excluded values?
  5. Fill the blank with directly, inversely, or neither. “The amount of time it takes to mow the lawn varies ________________ with the size of the lawn mower.”

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