What if you had two rational expressions like \begin{align*}\frac{x + 5}{x}\end{align*} and \begin{align*}\frac{x^2 + 6x + 5}{x - 2}\end{align*} and you wanted to divide them? How could you do so such that the answer were in simplest terms? After completing this Concept, you'll be able to divide rational expressions like this one.

### Watch This

CK-12 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*}. 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 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^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:

\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:

\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**

#### 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^3-9x\end{align*}, where \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 \begin{align*}x\end{align*} bowls of cereal, is \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**

\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 \begin{align*}\frac{3}{8}\end{align*} of an hour.**

CK-12 Foundation: 1209 Dividing Rational Expressions

### 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^2-15x}{2x^2+3x-14} \div \frac{x^2-25}{2x^2+13x+21}\end{align*}.*

**Solution**

\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*}

### Practice

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

- \begin{align*}2xy \div \frac{2x^2}{y}\end{align*}
- \begin{align*}\frac{2x^3}{y} \div 3x^2\end{align*}
- \begin{align*}\frac{3x+6}{y-4} \div \frac{3y+9}{x-1}\end{align*}
- \begin{align*}\frac{x^2}{x-1} \div \frac{x}{x^2+x-2}\end{align*}
- \begin{align*}\frac{a^2+2ab+b^2}{ab^2-a^2b} \div (a+b)\end{align*}
- \begin{align*}\frac{3-x}{3x-5} \div \frac{x^2-9}{2x^2-8x-10}\end{align*}
- \begin{align*}\frac{x^2-25}{x+3} \div (x-5)\end{align*}
- \begin{align*}\frac{2x+1}{2x-1} \div \frac{4x^2-1}{1-2x}\end{align*}
- \begin{align*}\frac{3x^2+5x-12}{x^2-9} \div \frac{3x-4}{3x+4}\end{align*}
- \begin{align*}\frac{x^2+x-12}{x^2+4x+4} \div \frac{x-3}{x+2}\end{align*}
- \begin{align*}\frac{x^4-16}{x^2-9} \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*}.