Dividing Rational Expressions, divide a Rational Expression by another one, can be complicated. In this blog post, you will learn how to divide rational expressions into a few simple steps.

## Related Topics

- How to Add and Subtract Rational Expressions
- How to Multiply Rational Expressions
- How to Solve Rational Equations
- How to Simplify Complex Fractions
- How to Graph Rational Expressions

## Method of Dividing Rational Expressions

- To divide rational expression, use the same method we use for dividing fractions. (Keep, Change, Flip)
- Keep the first rational expression, change the division sign to multiplication, and flip the numerator and denominator of the second rational expression. Then, multiply numerators and multiply denominators. Simplify as needed.

## Examples

### Dividing Rational Expressions – Example 1:

\(\frac{x+2}{3x}÷\frac{x^2+5x+6}{3x^2+3x}\)=

**Solution:**

Use fractions division rule: \(\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}×\frac{d}{c}=\frac{a×d}{b×c}\)

\(\frac{x+2}{3x}÷\frac{x^2+5x+6}{3x^2+3x}=\frac{x+2}{3x}×\frac{3x^2+3x}{x^2+5x+6}=\frac{(x+2)(3x^2+3x)}{(3x)(x^2+5x+6)}\)

Now, factorize the expressions \(3x^2+3x\) and \((x^2+5x+6)\).

Then: \(3x^2+3x=3x(x+1)\) and \(x^2+5x+6=(x+2)(x+3)\)

Simplify: \(\frac{(x+2)(3x^2+3x)}{(3x)(x^2+5x+6)} =\frac{(x+2)(3x)(x+1)}{(3x)(x+2)(x+3)}\), cancel common factors. Then: \(\frac{(x+2)(3x)(x+1)}{(3x)(x+2)(x+3)}=\frac{x+1}{x+3}\)

### Dividing Rational Expressions – Example 2:

\(\frac{5x}{x + 3}÷\frac{x}{2x + 6}\)=

**Solution:**

Use fractions division rule: \(\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}×\frac{d}{c}=\frac{a×d}{b×c}\).

Then: \(\frac{5x}{x + 3}÷\frac{x}{2x + 6}=\frac{5x}{x + 3}×\frac{2x + 6}{x}=\frac{5x(2x + 6)}{x(x+3)}=\frac{5x×2(x+3)}{x(x+3)}\)

Cancel common factor: \(\frac{5x×2(x+3)}{x(x+3)}=\frac{10x(x+3)}{x(x+3)}=10\)

### Dividing Rational Expressions – Example 3:

\(\frac{2x}{5}÷\frac{8}{7}=\)

**Solution:**

\(\frac{2x}{5}÷\frac{8}{7}=\frac{\frac{2x}{5}}{\frac{8}{7}}\) , Use Divide fractions rules: \(\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a . d}{b . c}\)

\(\frac{\frac{2x}{5}}{\frac{8}{7}}=\frac{(2x)×7}{8×5}=\frac{14x}{40}=\frac{7x}{20}\)

### Dividing Rational Expressions – Example 4:

\(\frac{6x}{x + 2}÷\frac{x}{6x + 12}\)=

**Solution:**

\(\frac{\frac{6x}{x + 2}}{\frac{x}{6x + 12}}\) , Use Divide fractions rules: \(\frac{(6x)(6x+12)}{(x)(x+2)}\)

Cancel common fraction: \(\frac{(6x)(6x+12)}{(x)(x+2)}=\frac{36(x+2)}{(x+2)}=36\)

## Exercises for Dividing Rational Expressions

### Divide Rational Expressions.

- \(\color{blue}{\frac{2x}{7}÷\frac{4}{3}=}\)
- \(\color{blue}{\frac{3}{5x}÷\frac{9}{2x}=}\)
- \(\color{blue}{\frac{7x}{x+6}÷\frac{2}{x+6}=}\)
- \(\color{blue}{\frac{20x^2}{x-1}÷\frac{4x}{x+2}=}\)
- \(\color{blue}{\frac{2x-3}{x+4}÷\frac{5}{6x+24}=}\)
- \(\color{blue}{\frac{x+5}{4}÷\frac{x^2-25}{8}=}\)

- \(\color{blue}{\frac{3x}{14}}\)
- \(\color{blue}{\frac{2}{15}}\)
- \(\color{blue}{\frac{7x}{2}}\)
- \(\color{blue}{\frac{5x(x+2)}{x-1}}\)
- \(\color{blue}{\frac{6(2x-3)}{5}}\)
- \(\color{blue}{\frac{2}{x-5}}\)