12.9: Division of Rational Expressions
What if you had two rational expressions like \begin{align*}\frac{x + 5}{x}\end{align*}
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CK12 Foundation: 1209S Dividing Rational Expressions
Watch this video for more examples of how to multiply and divide rational expressions.
RobiChaudd: Multiply or divide rational expressions
Guidance
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 \begin{align*}\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}\end{align*}
Example A
Divide \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:
\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*}
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 B
Divide \begin{align*}\frac{9x^24}{2x2} \div (21x^22x8)\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:
\begin{align*}\frac{9x^24}{2x2} \div \frac{21x^22x8}{1} = \frac{9x^24}{2x2} \cdot \frac{1}{21x^22x8}\end{align*}
Then factor and solve:
\begin{align*}\frac{9x^24}{2x2} \cdot \frac{1}{21x^22x8} = \frac{(3x2)(3x+2)}{2(x1)} \cdot \frac{1}{(3x2)(7x+4)} = \frac{(3x+2)}{2(x1)} \cdot \frac{1}{(7x+4)} = \frac{3x+2}{14x^26x8}\end{align*}
Solve Applications Involving Multiplication and Division of Rational Expressions
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 \begin{align*}x^39x\end{align*}
Solution
\begin{align*}\text{time} = \frac{\text{distance}}{\text{speed}}\!\\
\\
\text{time} = \frac{3x^29x}{x^39x} = \frac{3x(x3)}{x(x^29)} = \frac{3x(x3)}{x(x+3)(x3)}\!\\
\\
\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 \begin{align*}\frac{3}{8}\end{align*}
Vocabulary
 When we multiply two fractions we multiply the numerators and denominators separately:
\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:
\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*}
Guided Practice
Divide \begin{align*}\frac{3x^215x}{2x^2+3x14} \div \frac{x^225}{2x^2+13x+21}\end{align*}
Solution
\begin{align*}\frac{3x^215x}{2x^2+3x14} \cdot \frac{2x^2+13x+21}{x^225} = \frac{3x(x5)}{(2x+7)(x2)} \cdot \frac{(2x+7)(x+3)}{(x5)(x+5)} = \frac{3x}{(x2)} \cdot \frac{(x+3)}{(x+5)} = \frac{3x^2+9x}{x^2+3x10}\end{align*}
Practice
Divide the rational functions and reduce the answer to lowest terms.

\begin{align*}2xy \div \frac{2x^2}{y}\end{align*}
2xy÷2x2y 
\begin{align*}\frac{2x^3}{y} \div 3x^2\end{align*}
2x3y÷3x2 
\begin{align*}\frac{3x+6}{y4} \div \frac{3y+9}{x1}\end{align*}
3x+6y−4÷3y+9x−1 
\begin{align*}\frac{x^2}{x1} \div \frac{x}{x^2+x2}\end{align*}
x2x−1÷xx2+x−2 
\begin{align*}\frac{a^2+2ab+b^2}{ab^2a^2b} \div (a+b)\end{align*}
a2+2ab+b2ab2−a2b÷(a+b) 
\begin{align*}\frac{3x}{3x5} \div \frac{x^29}{2x^28x10}\end{align*}
3−x3x−5÷x2−92x2−8x−10 
\begin{align*}\frac{x^225}{x+3} \div (x5)\end{align*}
x2−25x+3÷(x−5) 
\begin{align*}\frac{2x+1}{2x1} \div \frac{4x^21}{12x}\end{align*}
2x+12x−1÷4x2−11−2x 
\begin{align*}\frac{3x^2+5x12}{x^29} \div \frac{3x4}{3x+4}\end{align*}
3x2+5x−12x2−9÷3x−43x+4 
\begin{align*}\frac{x^2+x12}{x^2+4x+4} \div \frac{x3}{x+2}\end{align*}
x2+x−12x2+4x+4÷x−3x+2  \begin{align*}\frac{x^416}{x^29} \div \frac{x^2+4}{x^2+6x+9}\end{align*}
 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?
 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 \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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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.Image Attributions
Here you'll learn how to divide a rational expression by a polynomial. You'll also learn how to solve realworld applications that involve multiplication and division of rational expressions.