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12.5: Multiplication and Division of Rational Expressions

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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 $a \neq 0, b \neq 0, c \neq 0, d \neq 0$,

$\frac{a}{b} \cdot \frac{c}{d}= \frac{ac}{bd}\\\frac{a}{b} \div \frac{c}{d} \rightarrow \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}$

Example: Multiply the following $\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2}$

Solution:

$\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2} \rightarrow \frac{4ab^3}{80a^2 b^8}$

Simplify exponents using methods learned in chapter 8.

$\frac{4ab^3}{80a^2 b^8}=\frac{1}{20ab^5}$

Example 1: Simplify $9c^2 \cdot \frac{4y^2}{21c^4}$.

Solution:

$9c^2 \cdot \frac{4y^2}{21c^4} \rightarrow \frac{9c^2}{1} \cdot \frac{4y^2}{21c^4}\\\frac{9c^2}{1} \cdot \frac{4y^2}{21c^4}=\frac{36c^2 y^2}{21c^4}\\\frac{36c^2 y^2}{21c^4}=\frac{12y^2}{7c^2}$

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 $\frac{4x+12}{3x^2} \cdot \frac{x}{x^2-9}$.

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

$\frac{4x+12}{3x^2} \cdot & \frac{x}{x^2-9} \rightarrow \frac{4(x+3)}{3x^2} \cdot \frac{x}{(x+3)(x-3)}\\& \frac{4\cancel{(x+3)}}{3x^{\cancel{2}}} \cdot \frac{\cancel{x}}{\cancel{(x+3)}(x-3)}\\& \frac{4}{3x} \cdot \frac{1}{x-3} \rightarrow \frac{4}{3x^2-9x}$

Example 1: Multiply $\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18}$.

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

$\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18} & \rightarrow \frac{(3x+2)(4x-3)}{(x+1)(x-1)} \cdot \frac{(x+1)(x+6)}{(4x-3)(x-6)}\\\frac{(3x+2)\cancel{(4x-3)}}{\cancel{(x+1)}(x-1)} \cdot \frac{\cancel{(x+1)}(x+6)}{\cancel{(4x-3)}(x-6)} & \rightarrow \frac{(3x+2)(x+6)}{(x-1)(x-6)}\\\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18} &= \frac{3x^2+20x+12}{x^2-7x+6}$

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 $a \neq 0, b \neq 0, c \neq 0, d \neq 0$,

$\frac{a}{b} \div \frac{c}{d} \rightarrow \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}$

Example: Simplify $\frac{9x^2-4}{2x-2} \div \frac{21x^2-2x-8}{1}$.

Solution:

$\frac{9x^2-4}{2x-2} \div \frac{21x^2-2x-8}{1} \rightarrow \frac{9x^2-4}{2x-2} \cdot \frac{1}{21x^2-2x-8}$

Repeat the process for multiplying rational expressions.

$\frac{9x^2-4}{2x-2} \cdot \frac{1}{21x^2-2x-8} & \rightarrow \frac{(3x-2)\cancel{(3x-2)}}{2(x-1)} \cdot \frac{1}{\cancel{(3x-2)}(7x+4)}\\\frac{9x^2-4}{2x-2} \div \frac{21x^2-2x-8}{1} &= \frac{3x-2}{14x^2-6x-8}$

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 $x^3-9x$, where $x$ is the number of bowls of cereal he had for breakfast $(1 \le x \le 6)$. Marciel’s training distance (in miles), if he eats $x$ bowls of cereal, is $3x^2-9x$. 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?

$\text{time} &= \frac{\text{distance}}{\text{speed}}\\\text{time} &= \frac{3x^2-9x}{x^3-9x}=\frac{3x(x-3)}{x(x^2-9)}=\frac{3x\cancel{(x-3)}}{x(x+3)\cancel{(x-3)}}\\\text{time} &= \frac{3}{x+3}\\\text{If} \ x &= 5, \text{then}\\\text{time} &= \frac{3}{5+3}=\frac{3}{8}$.

Marciel will run for $\frac{3}{8}$ 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.

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

1. $\frac{x^3}{2y^3} \cdot \frac{2y^2}{x}$
2. $2xy \div \frac{2x^2}{y}$
3. $\frac{2x}{y^2} \cdot \frac{4y}{5x}$
4. $2xy \cdot \frac{2y^2}{x^3}$
5. $\frac{4y^2-1}{y^2-9} \cdot \frac{y-3}{2y-1}$
6. $\frac{6ab}{a^2} \cdot \frac{a^3b}{3b^2}$
7. $\frac{x^2}{x-1} \div \frac{x}{x^2+x-2}$
8. $\frac{33a^2}{-5} \cdot \frac{20}{11a^3}$
9. $\frac{a^2+2ab+b^2}{ab^2-a^2b} \div (a+b)$
10. $\frac{2x^2+2x-24}{x^2+3x} \cdot \frac{x^2+x-6}{x+4}$
11. $\frac{3-x}{3x-5} \div \frac{x^2-9}{2x^2-8x-10}$
12. $\frac{x^2-25}{x+3} \div (x-5)$
13. $\frac{2x+1}{2x-1} \div \frac{4x^2-1}{1-2x}$
14. $\frac{x}{x-5} \cdot \frac{x^2-8x+15}{x^2-3x}$
15. $\frac{3x^2+5x-12}{x^2-9} \div \frac{3x-4}{3x+4}$
16. $\frac{5x^2+16x+3}{36x^2-25} \cdot (6x^2+5x)$
17. $\frac{x^2+7x+10}{x^2-9} \cdot \frac{x^2-3x}{3x^2+4x-4}$
18. $\frac{x^2+x-12}{x^2+4x+4} \div \frac{x-3}{x+2}$
19. $\frac{x^4-16}{x^2-9} \div \frac{x^2+4}{x^2+6x+9}$
20. $\frac{x^2+8x+16}{7x^2+9x+2} \div \frac{7x+2}{x^2+4x}$
21. Maria’s recipe asks for $2 \frac{1}{2} \ \text{times}$ more flour than sugar. How many cups of flour should she mix in if she uses $3 \frac{1}{3} \ \text{cups}$ 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 $I=\frac{V}{R_c}$. The total resistance for resistors placed in parallel is given by $\frac{1}{R_{tot}}=\frac{1}{R_1}+\frac{1}{R_2}$. Write the formula for the electric current in term of the component resistances: $R_1$ and $R_2$.

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: $-24 \ge |2r+3|$.
4. What is true of any line parallel to $5x+9y=-36$?
5. Solve for $d: 3+5d=-d-(3x-3)$.
6. Graph and determine the domain and range: $y-9=-x^2-5x$.
7. Rewrite in vertex form by completing the square. Identify the vertex: $y^2-16y+3=4$.

Quick Quiz

1. $h$ is inversely proportional to $t$. If $t=-0.05153$ when $h=-16$, find $t$ when $h=1.45$.
2. Use $f(x)=\frac{-5}{x^2-25}$ 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 $\frac{8c^4+12c^2-22c+1}{4}$.
4. Simplify $\frac{10a^2-30a}{a-3}$. 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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Feb 22, 2012

Aug 21, 2014