# 12.6: Division of Rational Expressions

**Basic**Created by: CK-12

**Practice**Division of Rational Expressions

Suppose that the distance traveled by a hot air balloon in miles can be represented by \begin{align*}8x^3-8x\end{align*}

### 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 \begin{align*}a \neq 0, b \neq 0, c \neq 0, d \neq 0\end{align*}

\begin{align*}\frac{a}{b} \div \frac{c}{d} \rightarrow \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}\end{align*}

#### Example A

*Simplify \begin{align*}\frac{9x^2-4}{2x-2} \div \frac{21x^2-2x-8}{1}\end{align*} 9x2−42x−2÷21x2−2x−81.*

**Solution:**

\begin{align*}\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}\end{align*}

Repeat the process for multiplying rational expressions.

\begin{align*}\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}\end{align*}

#### Example B

*Simplify \begin{align*}\frac{x^2+3x-10}{5x+15} \div \frac{x-2}{x^2+2x-3}\end{align*} x2+3x−105x+15÷x−2x2+2x−3.*

**Solution:**

\begin{align*}\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}\end{align*}

Repeat the process for multiplying rational expressions.

\begin{align*}\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}\end{align*}

**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 \begin{align*}x^3-9x\end{align*}

**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\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}\end{align*}

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

### Guided Practice

*Simplify \begin{align*} \frac{1}{5x^2-30x+40} \div \frac{3x-6}{2x^2-8x}\end{align*}.*

**Solution:**

\begin{align*} \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} \end{align*}

### 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.

- \begin{align*}2xy \div \frac{2x^2}{y}\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*}
- \begin{align*}\frac{x^2+8x+16}{7x^2+9x+2} \div \frac{7x+2}{x^2+4x}\end{align*}
- Maria’s recipe asks for \begin{align*}2 \frac{1}{2} \ \text{times}\end{align*} more flour than sugar. How many cups of flour should she mix in if she uses \begin{align*}3 \frac{1}{3} \ \text{cups}\end{align*} 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 terms of the component resistances: \begin{align*}R_1\end{align*} and \begin{align*}R_2\end{align*}.

#### Quick Quiz

- \begin{align*}h\end{align*} is inversely proportional to \begin{align*}t\end{align*}. If \begin{align*}t=-0.05153\end{align*} when \begin{align*}h=-16\end{align*}, find \begin{align*}t\end{align*} when \begin{align*}h=1.45\end{align*}.
- Use \begin{align*}f(x)=\frac{-5}{x^2-25}\end{align*} for the following questions.
- Find the excluded values.
- Determine the vertical asymptotes.
- Sketch a graph of this function.
- Determine its domain and range.

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

### Notes/Highlights Having trouble? Report an issue.

Color | Highlighted Text | Notes | |
---|---|---|---|

Show More |

Term | Definition |
---|---|

rational function |
A ratio of two polynomials (a polynomial divided by another polynomial). The formal definition is: . |

reciprocal |
The of a nonzero rational expression is .reciprocal |

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 find the quotient of two rational expressions.