12.5: Multiplication and Division of Rational Expressions
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
Example: Multiply the following
Solution:
Simplify exponents using methods learned in chapter 8.
Example 1: Simplify
Solution:
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
Solution: Factor all pieces of these rational expressions and reduce before multiplying.
& \frac{4\cancel{(x+3)}}{3x^{\cancel{2}}} \cdot \frac{\cancel{x}}{\cancel{(x+3)}(x3)}\\
& \frac{4}{3x} \cdot \frac{1}{x3} \rightarrow \frac{4}{3x^29x}
Example 1: Multiply
Solution: Factor all pieces, reduce, and then multiply.
\frac{(3x+2)\cancel{(4x3)}}{\cancel{(x+1)}(x1)} \cdot \frac{\cancel{(x+1)}(x+6)}{\cancel{(4x3)}(x6)} & \rightarrow \frac{(3x+2)(x+6)}{(x1)(x6)}\\
\frac{12x^2x6}{x^21} \cdot \frac{x^2+7x+6}{4x^227x+18} &= \frac{3x^2+20x+12}{x^27x+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
Example: Simplify
Solution:
Repeat the process for multiplying rational expressions.
\frac{9x^24}{2x2} \div \frac{21x^22x8}{1} &= \frac{3x2}{14x^26x8}
RealLife 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
\text{time} &= \frac{3x^29x}{x^39x}=\frac{3x(x3)}{x(x^29)}=\frac{3x\cancel{(x3)}}{x(x+3)\cancel{(x3)}}\\
\text{time} &= \frac{3}{x+3}\\
\text{If} \ x &= 5, \text{then}\\
\text{time} &= \frac{3}{5+3}=\frac{3}{8}
Marciel will run for
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.
CK12 Basic Algebra: Multiplying and Dividing Rational Expressions (9:19)
In 1–20, perform the indicated operation and reduce the answer to lowest terms

x32y3⋅2y2x 
2xy÷2x2y 
2xy2⋅4y5x 
2xy⋅2y2x3 
4y2−1y2−9⋅y−32y−1 
6aba2⋅a3b3b2 
x2x−1÷xx2+x−2 
33a2−5⋅2011a3 
a2+2ab+b2ab2−a2b÷(a+b) 
2x2+2x−24x2+3x⋅x2+x−6x+4 
3−x3x−5÷x2−92x2−8x−10 
x2−25x+3÷(x−5) 
2x+12x−1÷4x2−11−2x 
xx−5⋅x2−8x+15x2−3x 
3x2+5x−12x2−9÷3x−43x+4 
5x2+16x+336x2−25⋅(6x2+5x) 
x2+7x+10x2−9⋅x2−3x3x2+4x−4 
x2+x−12x2+4x+4÷x−3x+2 
x4−16x2−9÷x2+4x2+6x+9 
x2+8x+167x2+9x+2÷7x+2x2+4x  Maria’s recipe asks for
212 times more flour than sugar. How many cups of flour should she mix in if she uses313 cups of sugar?  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?
 Ohm’s Law states that in an electrical circuit
I=VRc . The total resistance for resistors placed in parallel is given by1Rtot=1R1+1R2 . Write the formula for the electric current in term of the component resistances:R1 andR2 .
Mixed Review
 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?
 A local nursery makes two types of fall arrangements. One arrangement uses eight mums and five blackeyed susans. The other arrangement uses six mums and 9 blackeyed susans. The nursery can use no more than 144 mums and 135 blackeyed 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?
 Solve for
r and graph the solution on a number line:−24≥2r+3 .  What is true of any line parallel to
5x+9y=−36 ?  Solve for
d:3+5d=−d−(3x−3) .  Graph and determine the domain and range:
y−9=−x2−5x .  Rewrite in vertex form by completing the square. Identify the vertex:
y2−16y+3=4 .
Quick Quiz

h is inversely proportional tot . Ift=−0.05153 whenh=−16 , findt whenh=1.45 .  Use
f(x)=−5x2−25 for the following questions. Find the excluded values.
 Determine the vertical asymptotes.
 Sketch a graph of this function.
 Determine its domain and range.
 Simplify
8c4+12c2−22c+14 .  Simplify
10a2−30aa−3 . What are its excluded values?  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.”