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# 12.5: Multiplying and Dividing Rational Expressions

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Multiply rational expressions involving monomials.
• Multiply rational expressions involving polynomials.
• Multiply a rational expression by a polynomial.
• Divide rational expressions involving polynomials.
• Divide a rational expression by a polynomial.
• Solve real-world problems involving multiplication and division of rational expressions.

## Introduction

The rules for multiplying and dividing rational expressions are the same as the rules for multiplying and dividing rational numbers, so let’s start by reviewing multiplication and division of fractions. When we multiply two fractions we multiply the numerators and denominators separately:

abcd=acbd\begin{align*}\frac{a}{b} \cdot \frac{c}{d}=\frac{a \cdot c}{b \cdot d}\end{align*}

When we divide two fractions, we replace the second fraction with its reciprocal and multiply, since that’s mathematically the same operation:

ab÷cd=abdc=adbc\begin{align*}\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a \cdot d}{b \cdot c}\end{align*}

## Multiply Rational Expressions Involving Monomials

Example 1

Multiply the following: a16b84b35a2\begin{align*}\frac{a}{16b^8} \cdot \frac{4b^3}{5a^2}\end{align*}.

Solution

Cancel common factors from the numerator and denominator. The common factors are 4, a\begin{align*}a\end{align*}, and b3\begin{align*}b^3\end{align*}. Canceling them out leaves 14b515a=120ab5\begin{align*}\frac{1}{4b^5} \cdot \frac{1}{5a} = \frac{1}{20ab^5}\end{align*}.

Example 2

Multiply 9x24y221x4\begin{align*}9x^2 \cdot \frac{4y^2}{21x^4}\end{align*}.

Solution

Rewrite the problem as a product of two fractions: 9x214y221x4\begin{align*}\frac{9x^2}{1} \cdot \frac{4y^2}{21x^4}\end{align*} Then cancel common factors from the numerator and denominator.

The common factors are 3 and x2\begin{align*}x^2\end{align*}. Canceling them out leaves 314y27x2=12y27x2\begin{align*}\frac{3}{1} \cdot \frac{4y^2}{7x^2} = \frac{12y^2}{7x^2}\end{align*}.

## Multiply Rational Expressions Involving Polynomials

When multiplying rational expressions involving polynomials, first we need to factor all polynomial expressions as much as we can. Then we follow the same procedure as before.

Example 3

Multiply 4x+123x2xx29\begin{align*}\frac{4x+12}{3x^2} \cdot \frac{x}{x^2-9}\end{align*}.

Solution

Factor all polynomial expressions as much as possible: 4(x+3)3x2x(x+3)(x3)\begin{align*}\frac{4(x+3)}{3x^2} \cdot \frac{x}{(x+3)(x-3)}\end{align*}

The common factors are x\begin{align*}x\end{align*} and (x+3)\begin{align*}(x + 3)\end{align*}. Canceling them leaves 43x1(x3)=43x(x3)=43x29x\begin{align*}\frac{4}{3x} \cdot \frac{1}{(x-3)} = \frac{4}{3x(x-3)} = \frac{4}{3x^2-9x}\end{align*}.

Example 4

Multiply 12x2x6x21x2+7x+64x227x+18\begin{align*}\frac{12x^2-x-6}{x^2-1} \cdot \frac{x^2+7x+6}{4x^2-27x+18}\end{align*}.

Solution

Factor polynomials: (3x+2)(4x3)(x+1)(x1)(x+1)(x+6)(4x3)(x6)\begin{align*}\frac{(3x+2)(4x-3)}{(x+1)(x-1)} \cdot \frac{(x+1)(x+6)}{(4x-3)(x-6)}\end{align*}.

The common factors are (x+1)\begin{align*}(x + 1)\end{align*} and (4x3)\begin{align*}(4x - 3)\end{align*}. Canceling them leaves (3x+2)(x1)(x+6)(x6)=(3x+2)(x+6)(x1)(x6)=3x2+20x+12x27x+6\begin{align*}\frac{(3x+2)}{(x-1)} \cdot \frac{(x+6)}{(x-6)} = \frac{(3x+2)(x+6)}{(x-1)(x-6)} = \frac{3x^2+20x+12}{x^2-7x+6}\end{align*}

## Multiply a Rational Expression by a Polynomial

When we multiply 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. We then proceed the same way as in the previous examples.

Example 5

Multiply 3x+184x2+19x5(x2+3x10)\begin{align*}\frac{3x+18}{4x^2+19x-5} \cdot (x^2+3x-10)\end{align*}.

Solution

Rewrite the expression as a product of fractions: 3x+184x2+19x5x2+3x101\begin{align*}\frac{3x+18}{4x^2+19x-5} \cdot \frac{x^2+3x-10}{1}\end{align*}

Factor polynomials: 3(x+6)(x+5)(4x1)(x2)(x+5)1\begin{align*}\frac{3(x+6)}{(x+5)(4x-1)} \cdot \frac{(x-2)(x+5)}{1}\end{align*}

The common factor is (x+5)\begin{align*}(x + 5)\end{align*}. Canceling it leaves 3(x+6)(4x1)(x2)1=3(x+6)(x2)(4x1)=3x2+12x364x1\begin{align*}\frac{3(x+6)}{(4x-1)} \cdot \frac{(x-2)}{1} = \frac{3(x+6)(x-2)}{(4x-1)} = \frac{3x^2+12x-36}{4x-1}\end{align*}

## Divide Rational Expressions Involving Polynomials

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 example.

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.

Example 6

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

Solution

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*}

Example 7

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*}.

Solution

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*}

## Divide 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.

Example 8

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

Solution

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*}

For more examples of how to multiply and divide rational expressions, watch the video at

.

## Solve Real-World Problems Involving Multiplication and Division of Rational Expressions

Example 9

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?

Solution

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.

## Review Questions

Perform the indicated operation and reduce the answer to lowest terms.

1. x32y32y2x\begin{align*}\frac{x^3}{2y^3} \cdot \frac{2y^2}{x}\end{align*}
2. 2xy÷2x2y\begin{align*}2xy \div \frac{2x^2}{y}\end{align*}
3. 2xy24y5x\begin{align*}\frac{2x}{y^2} \cdot \frac{4y}{5x}\end{align*}
4. 2x3y÷3x2\begin{align*}\frac{2x^3}{y} \div 3x^2\end{align*}
5. 2xy2y2x3\begin{align*}2xy \cdot \frac{2y^2}{x^3}\end{align*}
6. 3x+6y4÷3y+9x1\begin{align*}\frac{3x+6}{y-4} \div \frac{3y+9}{x-1}\end{align*}
7. 4y21y29y32y1\begin{align*}\frac{4y^2-1}{y^2-9} \cdot \frac{y-3}{2y-1}\end{align*}
8. 6aba2a3b3b2\begin{align*}\frac{6ab}{a^2} \cdot \frac{a^3b}{3b^2}\end{align*}
9. x2x1÷xx2+x2\begin{align*}\frac{x^2}{x-1} \div \frac{x}{x^2+x-2}\end{align*}
10. 33a252011a3\begin{align*}\frac{33a^2}{-5} \cdot \frac{20}{11a^3}\end{align*}
11. a2+2ab+b2ab2a2b÷(a+b)\begin{align*}\frac{a^2+2ab+b^2}{ab^2-a^2b} \div (a+b)\end{align*}
12. \begin{align*}\frac{2x^2+2x-24}{x^2+3x} \cdot \frac{x^2+x-6}{x+4}\end{align*}
13. \begin{align*}\frac{3-x}{3x-5} \div \frac{x^2-9}{2x^2-8x-10}\end{align*}
14. \begin{align*}\frac{x^2-25}{x+3} \div (x-5)\end{align*}
15. \begin{align*}\frac{2x+1}{2x-1} \div \frac{4x^2-1}{1-2x}\end{align*}
16. \begin{align*}\frac{x}{x-5} \cdot \frac{x^2-8x+15}{x^2-3x}\end{align*}
17. \begin{align*}\frac{3x^2+5x-12}{x^2-9} \div \frac{3x-4}{3x+4}\end{align*}
18. \begin{align*}\frac{5x^2+16x+3}{36x^2-25} \cdot (6x^2+5x)\end{align*}
19. \begin{align*}\frac{x^2+7x+10}{x^2-9} \cdot \frac{x^2-3x}{3x^2+4x-4}\end{align*}
20. \begin{align*}\frac{x^2+x-12}{x^2+4x+4} \div \frac{x-3}{x+2}\end{align*}
21. \begin{align*}\frac{x^4-16}{x^2-9} \div \frac{x^2+4}{x^2+6x+9}\end{align*}
22. \begin{align*}\frac{x^2+8x+16}{7x^2+9x+2} \cdot \frac{7x+2}{x^2+4x}\end{align*}
23. Maria’s recipe asks for \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 \begin{align*}3 \frac{1}{3}\end{align*} cups of sugar?
24. 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?
25. Ohm’s Law states that in an electrical circuit \begin{align*}I = \frac{V}{R_c}\end{align*}. The total resistance for resistors placed in parallel is given by: \begin{align*}\frac{1}{R_{tot}} = \frac{1}{R_1} + \frac{1}{R_2}\end{align*}. Write the formula for the electric current in term of the component resistances: \begin{align*}R_1\end{align*} and \begin{align*}R_2\end{align*}.

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