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# 12.9: Division of Rational Expressions

Difficulty Level: At Grade Created by: CK-12
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Practice Division of Rational Expressions
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### Division of Rational Expressions

Just as with ordinary fractions, we first rewrite the division problem as a multiplication problem and then proceed with the multiplication as outlined in the previous section.

Note: Remember that ab÷cd=abdc\begin{align*}\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}\end{align*}. The first fraction remains the same and you take the reciprocal of the second fraction. Do not fall into the common trap of flipping the first fraction.

Divide 4x215÷6x5\begin{align*}\frac{4x^2}{15} \div \frac{6x}{5}\end{align*}.

First convert into a multiplication problem by flipping the second fraction and then simplify as usual:

4x215÷6x5=4x21556x=2x313=2x9\begin{align*}\frac{4x^2}{15} \div \frac{6x}{5} = \frac{4x^2}{15} \cdot \frac{5}{6x} = \frac{2x}{3} \cdot \frac{1}{3} = \frac{2x}{9}\end{align*}

Dividing a Rational Expression by a Polynomial

When we divide a rational expression by a whole number or a polynomial, we can write the whole number (or polynomial) as a fraction with denominator equal to one, and then proceed the same way as in the previous examples.

Divide 9x242x2÷(21x22x8)\begin{align*}\frac{9x^2-4}{2x-2} \div (21x^2-2x-8)\end{align*}.

Rewrite the expression as a division of fractions, and then convert into a multiplication problem by taking the reciprocal of the divisor:

9x242x2÷21x22x81=9x242x2121x22x8\begin{align*}\frac{9x^2-4}{2x-2} \div \frac{21x^2-2x-8}{1} = \frac{9x^2-4}{2x-2} \cdot \frac{1}{21x^2-2x-8}\end{align*}

Then factor and solve:

9x242x2121x22x8=(3x2)(3x+2)2(x1)1(3x2)(7x+4)=(3x+2)2(x1)1(7x+4)=3x+214x26x8\begin{align*}\frac{9x^2-4}{2x-2} \cdot \frac{1}{21x^2-2x-8} = \frac{(3x-2)(3x+2)}{2(x-1)} \cdot \frac{1}{(3x-2)(7x+4)} = \frac{(3x+2)}{2(x-1)} \cdot \frac{1}{(7x+4)} = \frac{3x+2}{14x^2-6x-8}\end{align*}

#### Solve Applications Involving Multiplication and Division of Rational Expressions

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\begin{align*}x^3-9x\end{align*}, where x\begin{align*}x\end{align*} is the number of bowls of cereal he had for breakfast. Marciel’s training distance (in miles), if he eats x\begin{align*}x\end{align*} bowls of cereal, is 3x29x\begin{align*}3x^2-9x\end{align*}. 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?

time=distancespeedtime=3x29xx39x=3x(x3)x(x29)=3x(x3)x(x+3)(x3)time=3x+3If x=5, thentime=35+3=38\begin{align*}\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(x-3)}{x(x+3)(x-3)}\!\\ \\ \text{time} = \frac{3}{x+3}\!\\ \\ \text{If} \ x = 5, \ \text{then}\!\\ \\ \text{time} = \frac{3}{5+3}=\frac{3}{8}\end{align*}

Marciel will run for 38\begin{align*}\frac{3}{8}\end{align*} of an hour.

### Example

#### Example 1

Divide 3x215x2x2+3x14÷x2252x2+13x+21\begin{align*}\frac{3x^2-15x}{2x^2+3x-14} \div \frac{x^2-25}{2x^2+13x+21}\end{align*}.

3x215x2x2+3x142x2+13x+21x225=3x(x5)(2x+7)(x2)(2x+7)(x+3)(x5)(x+5)=3x(x2)(x+3)(x+5)=3x2+9xx2+3x10\begin{align*}\frac{3x^2-15x}{2x^2+3x-14} \cdot \frac{2x^2+13x+21}{x^2-25} = \frac{3x(x-5)}{(2x+7)(x-2)} \cdot \frac{(2x+7)(x+3)}{(x-5)(x+5)} = \frac{3x}{(x-2)} \cdot \frac{(x+3)}{(x+5)} = \frac{3x^2+9x}{x^2+3x-10}\end{align*}

### Review

Divide the rational functions and reduce the answer to lowest terms.

1. 2xy÷2x2y\begin{align*}2xy \div \frac{2x^2}{y}\end{align*}
2. 2x3y÷3x2\begin{align*}\frac{2x^3}{y} \div 3x^2\end{align*}
3. 3x+6y4÷3y+9x1\begin{align*}\frac{3x+6}{y-4} \div \frac{3y+9}{x-1}\end{align*}
4. x2x1÷xx2+x2\begin{align*}\frac{x^2}{x-1} \div \frac{x}{x^2+x-2}\end{align*}
5. a2+2ab+b2ab2a2b÷(a+b)\begin{align*}\frac{a^2+2ab+b^2}{ab^2-a^2b} \div (a+b)\end{align*}
6. 3x3x5÷x292x28x10\begin{align*}\frac{3-x}{3x-5} \div \frac{x^2-9}{2x^2-8x-10}\end{align*}
7. x225x+3÷(x5)\begin{align*}\frac{x^2-25}{x+3} \div (x-5)\end{align*}
8. 2x+12x1÷4x2112x\begin{align*}\frac{2x+1}{2x-1} \div \frac{4x^2-1}{1-2x}\end{align*}
9. 3x2+5x12x29÷3x43x+4\begin{align*}\frac{3x^2+5x-12}{x^2-9} \div \frac{3x-4}{3x+4}\end{align*}
10. x2+x12x2+4x+4÷x3x+2\begin{align*}\frac{x^2+x-12}{x^2+4x+4} \div \frac{x-3}{x+2}\end{align*}
11. x416x29÷x2+4x2+6x+9\begin{align*}\frac{x^4-16}{x^2-9} \div \frac{x^2+4}{x^2+6x+9}\end{align*}
12. Maria’s recipe asks for 212\begin{align*}2 \frac{1}{2}\end{align*} times more flour than sugar. How many cups of flour should she mix in if she uses 313\begin{align*}3 \frac{1}{3}\end{align*} cups of sugar?
13. 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?
14. Ohm’s Law states that in an electrical circuit I=VRc\begin{align*}I = \frac{V}{R_c}\end{align*}. The total resistance for resistors placed in parallel is given by: 1Rtot=1R1+1R2\begin{align*}\frac{1}{R_{tot}} = \frac{1}{R_1} + \frac{1}{R_2}\end{align*}. Write the formula for the electric current in terms of the component resistances: R1\begin{align*}R_1\end{align*} and R2\begin{align*}R_2\end{align*}.

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### Vocabulary Language: English

Least Common Denominator

The least common denominator or lowest common denominator of two fractions is the smallest number that is a multiple of both of the original denominators.

Least Common Multiple

The least common multiple of two numbers is the smallest number that is a multiple of both of the original numbers.

Rational Expression

A rational expression is a fraction with polynomials in the numerator and the denominator.

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