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Division of Rational Expressions

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Suppose that the distance traveled by a hot air balloon in miles can be represented by $8x^3-8x$ , while the speed of the hot air balloon in miles per hour can be represented by $x^2+x$ . Would you be able to find the time it took for the hot air balloon to cover the distance given? Could you evaluate the expression that you found for $x=2$ ? In this Concept, you'll learn about the division of rational expressions so that you can solve problems like this one.

Guidance

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 A

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.

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

Example B

Simplify $\frac{x^2+3x-10}{5x+15} \div \frac{x-2}{x^2+2x-3}$ .

Solution:

$\frac{x^2+3x-10}{5x+15} \div \frac{x-2}{x^2+2x-3} \rightarrow \frac{x^2+3x-10}{5x+15} \cdot \frac{x^2+2x-3}{x-2}$

Repeat the process for multiplying rational expressions.

$\frac{x^2+3x-10}{5x+15} \cdot \frac{x^2+2x-3}{x-2} & \rightarrow \frac{(x+5)(x-2)}{5(x+3)} \cdot \frac{x-2}{(x+3)(x-1)}\\\frac{(x+5)\cancel{(x-2)}}{5\cancel{(x+3)}} \cdot \frac{\cancel{x-2}}{(\cancel{x+3})(x-1)}&=\frac{x+5}{5} \cdot \frac{1}{(x-1)}=\frac{x+5}{5x-5}\\\frac{x^2+3x-10}{5x+15} \div \frac{x-2}{x^2+2x-3} &= \frac{x+5}{5x-5}$

Real-Life Application of Rational Functions

Example C

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?

Solution:

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

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

Guided Practice

Simplify $\frac{1}{5x^2-30x+40} \div \frac{3x-6}{2x^2-8x}$ .

Solution:

$\frac{1}{5x^2-30x+40} \div \frac{3x-6}{2x^2-8x} &= \frac{1}{5x^2-30x+40} \cdot \frac{2x^2-8x}{3x-6}\\ &= \frac{1}{5(x-2)(x-4)} \cdot \frac{2x(x-4)}{3(x-2)}\\ &= \frac{1}{5(x-2) \cancel{(x-4)}} \cdot \frac{2x \cancel{(x-4)}}{3(x-2)}\\&=\frac{2x}{5(x-2)^2}$

Practice

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 1–10, perform the indicated operation and reduce the answer to lowest terms.

1. $2xy \div \frac{2x^2}{y}$
2. $\frac{x^2}{x-1} \div \frac{x}{x^2+x-2}$
3. $\frac{a^2+2ab+b^2}{ab^2-a^2b} \div (a+b)$
4. $\frac{3-x}{3x-5} \div \frac{x^2-9}{2x^2-8x-10}$
5. $\frac{x^2-25}{x+3} \div (x-5)$
6. $\frac{2x+1}{2x-1} \div \frac{4x^2-1}{1-2x}$
7. $\frac{3x^2+5x-12}{x^2-9} \div \frac{3x-4}{3x+4}$
8. $\frac{x^2+x-12}{x^2+4x+4} \div \frac{x-3}{x+2}$
9. $\frac{x^4-16}{x^2-9} \div \frac{x^2+4}{x^2+6x+9}$
10. $\frac{x^2+8x+16}{7x^2+9x+2} \div \frac{7x+2}{x^2+4x}$
11. 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?
12. 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?
13. 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 terms of the component resistances: $R_1$ and $R_2$ .

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.
1. Simplify $\frac{8c^4+12c^2-22c+1}{4}$ .
2. Simplify $\frac{10a^2-30a}{a-3}$ . What are its excluded values?
3. 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.”