# Division of Rational Expressions

## Invert, cancel, multiply, and reduce fractions with variables in the denominator

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Dividing Rational Expressions

The area of a rectangle is \begin{align*}\frac{12x^2yz^3}{5xy^2z}\end{align*}. The length of the rectangle is \begin{align*}\frac{2xy}{z^2}\end{align*}. What is the width of the rectangle?

### Dividing Rational Expressions

Dividing rational expressions requires one more step than multiplying them does. Recall that when you divide fractions, you need to flip the second fraction and change the problem to multiplication. The same rule applies to dividing rational expressions.

Divide the following rational expressions.

1. \begin{align*}\frac{5a^3b^4}{12ab^8} \div \frac{15b^6}{8a^6}\end{align*}

Flip the second fraction, change the \begin{align*}\div\end{align*} sign to multiplication, and simplify.

\begin{align*}\frac{5a^3b^4}{12ab^8} \div \frac{15b^6}{8a^6} = \frac{5a^3b^4}{12ab^8} \cdot \frac{8a^6}{15b^6} = \frac{40a^9b^4}{180ab^{14}} = \frac{2a^8}{9b^{10}}\end{align*}

1. \begin{align*}\frac{x^4-3x^2-4}{2x^2+x-10} \div \frac{x^3-3x^2+x-3}{x-2}\end{align*}

Flip the second fraction, change the \begin{align*}\div\end{align*} sign to multiplication and simplify.

\begin{align*}\frac{x^4-3x^2-4}{2x^2+x-10} \div \frac{x^3-3x^2+x-3}{x-2} &= \frac{{\color{red}x^4-3x^2-4}}{2x^2+x-10} \cdot \frac{x-2}{{\color{blue}x^3-3x^2+x-3}} \\ &= \frac{(x^2-4)(x^2+1)}{(2x-5)(x+2)} \cdot \frac{x-2}{(x^2+1)(x-3)} \\ &= \frac{(x-2)\cancel{(x+2)} \cancel{(x^2+1)}}{(2x-5)\cancel{(x+2)}} \cdot \frac{x-2}{\cancel{(x^2+1)}(x-3)} \\ &= \frac{(x-2)^2}{(2x-5)(x-3)}\end{align*}

Now perform the indicated operations: \begin{align*}\frac{{\color{blue}x^3-8}}{x^2-6x+9} \div (x^2+3x-10) \cdot \frac{x^2+x-12}{x^2+11x+30}\end{align*}.

Flip the second term, factor, and cancel (remember \begin{align*}x^3 -8\end{align*} is a difference of cubes).

\begin{align*}\frac{x^3-8}{x^2-6x+9} \div (x^2+3x-10) \cdot \frac{x^2+x-12}{x^2+11x+30} &= \frac{x^3-8}{x^2-6x+9} \cdot \frac{1}{x^2+3x-10} \cdot \frac{x^2+2x-15}{x^2+11x+30} \\ &= \frac{\cancel{(x-2)}(x^2+2x+4)}{\cancel{(x-3)}(x-3)} \cdot \frac{1}{\cancel{(x-2)} \cancel{(x+5)}} \cdot \frac{\cancel{(x+5)} \cancel{(x-3)}}{(x+5)(x+6)} \\ &= \frac{x^2+2x+4}{(x-3)(x+5)(x+6)}\end{align*}

### Examples

#### Example 1

Earlier, you were asked to find the width of a rectangle.

To find the width, divide the area by the length and simplify.

\begin{align*}\frac{12x^2yz^3}{5xy^2z} \div \frac{2xy}{z^2}\\ \frac{12x^2yz^3}{5xy^2z} \cdot \frac{z^2}{2xy}\\ \frac{12x^2yz^5}{10x^2y^3z}\\ \frac{6z^4}{5y^2}\end{align*}

Therefore, the width of the rectangle is \begin{align*}\frac{6z^4}{5y^2}\end{align*}.

Perform the indicated operations.

#### Example 2

\begin{align*}\frac{a^5b^3c}{6a^2c^9} \div \frac{2a^7b^{11}}{24c^2}\end{align*}

Invert the second fraction and simplify:

\begin{align*}\frac{a^5b^3c}{6a^2c^9} \div \frac{2a^7b^{11}}{24c^2} = \frac{a^5b^3c}{6a^2c^9} \cdot \frac{24c^2}{2a^7b^{11}} = \frac{24a^5b^3c^3}{12a^9b^{11}c^9} = \frac{2}{a^4b^8c^6}\end{align*}

#### Example 3

\begin{align*}\frac{x^2+12x-45}{x^2-5x+6} \div \frac{x^2+17x+30}{x^4-16}\end{align*}

Invert the second fraction and simplify:
\begin{align*}\frac{x^2+12x-45}{x^2-5x+6} \div \frac{x^2+17x+30}{x^4-16} &= \frac{x^2+12x-45}{x^2-5x+6} \cdot \frac{x^4-16}{x^2+17x+30} \\ &= \frac{\cancel{(x+15)}\cancel{(x-3)}}{\cancel{(x-3)} \cancel{(x-2)}} \cdot \frac{(x^2+4)\cancel{(x-2)} \cancel{(x+2)}}{\cancel{(x+15)} \cancel{(x+2)}} \\ &= x^2+4\end{align*}

#### Example 4

\begin{align*}(x^3+2x^2-9x-18) \div \frac{x^2+11x+24}{x^2-11x-24} \div \frac{x^2-6x-16}{x^2+5x-24}\end{align*}

Write the first term over one, invert the second and third fractions, and simplify:
\begin{align*}(x^3+2x^2-9x-18) \div \frac{x^2+11x+24}{x^2-11x+24} \div \frac{x^2-6x-16}{x^2+5x-24} &= \frac{x^3+2x^2-9x-18}{1} \cdot \frac{x^2-11x+24}{x^2+11x+24} \cdot \frac{x^2+5x-24}{x^2-6x-16} \\ &= \frac{(x-3) \cancel{(x+3)} \cancel{(x+2)}}{1} \cdot \frac{\cancel{(x-8)}(x-3)}{\cancel{(x+8)}\cancel{(x+3)}} \cdot \frac{\cancel{(x+8)}(x-3)}{\cancel{(x-8)} \cancel{(x+2)}} \\ &=(x-3)^2\end{align*}

### Review

1. \begin{align*}\frac{6a^4b^3}{8a^3b^6} \div \frac{3a^5}{4a^3b^4}\end{align*}
2. \begin{align*}\frac{12x^5y}{xy^4} \div \frac{18x^3y^6}{3x^2y^3}\end{align*}
3. \begin{align*}\frac{16x^3y^9z^3}{15x^5y^2z} \div \frac{42xy^7z^2}{45x^2yz^5}\end{align*}
4. \begin{align*}\frac{x^2+2x-3}{x^2-3x+2} \div \frac{x^2+3x}{4x-8}\end{align*}
5. \begin{align*}\frac{x^2-2x-3}{x^2+6x+5} \div \frac{4x-12}{x^2+8x+15}\end{align*}
6. \begin{align*}\frac{x^2+6x+2}{12-3x} \div \frac{6x^2-13x-5}{x^2-4x}\end{align*}
7. \begin{align*}\frac{x^2-5x}{x^2+x-6} \div \frac{x^2-2x-15}{x^3+3x^2-4x-12}\end{align*}
8. \begin{align*}\frac{3x^3-3x^2-6x}{2x^2+15x-8} \div \frac{6x^2+18x-60}{2x^2+9x-5}\end{align*}
9. \begin{align*}\frac{x^3+27}{x^2+5x-14} \div \frac{x^2-x-12}{2x^2+2x-40} \div \frac{1}{x-2}\end{align*}
10. \begin{align*}\frac{x^2+2x-15}{2x^3+7x^2-4x} \div (5x+3) \div \frac{21-10x+x^2}{5x^3+23x^2+12x}\end{align*}

We all know that when you divide fractions, you take the second fraction, flip it, and change it to a multiplication problem. But, do you know why? Let's investigate the why here.

1. What is \begin{align*}6 \div 2\end{align*}?
2. What about \begin{align*}\frac{1 \div 1}{6 \div 2}\end{align*}?
3. Is the problem above the same as \begin{align*}\frac{1}{6} \div \frac{1}{2}\end{align*}? Why or why not?

Let's take a different approach. Let's write a division problem as a huge fraction: \begin{align*}\frac{\frac{30}{52}}{\frac{15}{13}}\end{align*}

1. We know we cannot have fractions in the denominator of another fraction. What would we have to multiply the denominator by to cancel it out?
2. Multiply the top and bottom from your answer in #14. What did you multiply by?

To see the Review answers, open this PDF file and look for section 9.9.

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### Vocabulary Language: English

TermDefinition
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.