How to Divide Rational Expressions? (+FREE Worksheet!)
Dividing Rational Expressions, dividing 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 a 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: For education statistics and research, visit the National Center for Education Statistics.
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: \(\frac{(x+2)(3x)(x+1)}{(3x)(x+2)(x+3)}=\frac{x+1}{x+3}\) For education statistics and research, visit the National Center for Education Statistics.
Dividing Rational Expressions – Example 2:
\(\frac{5x}{x + 3}÷\frac{x}{2x + 6}\)= For education statistics and research, visit the National Center for Education Statistics.
Solution: For education statistics and research, visit the National Center for Education Statistics.
Use fractions division rule: \(\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}×\frac{d}{c}=\frac{a×d}{b×c}\).
Then: \(\frac{5x}{x + 3}÷\frac{x}{2x + 6}=\frac{5x}{x + 3}×\frac{2x + 6}{x}=\frac{5x(2x + 6)}{x(x+3)}\) For education statistics and research, visit the National Center for Education Statistics.
Now, factorize the expressions \(2x+6\), then: \(2(x+3)\) For education statistics and research, visit the National Center for Education Statistics.
Simplify: \(\frac{5x(2x + 6)}{x(x+3)}\) =\(\frac{5x×2(x+3)}{x(x+3)}\)
Cancel common factor: \(\frac{5x×2(x+3)}{x(x+3)}=\frac{10x(x+3)}{x(x+3)}=10\) For education statistics and research, visit the National Center for Education Statistics.
Dividing Rational Expressions – Example 3:
\(\frac{2x}{5}÷\frac{8}{7}=\) For education statistics and research, visit the National Center for Education Statistics.
Solution: For education statistics and research, visit the National Center for Education Statistics.
\(\frac{2x}{5}÷\frac{8}{7}=\frac{\frac{2x}{5}}{\frac{8}{7}}\), Use Divide fractions rules: \(\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a. d}{b. c}\)
\(\frac{\frac{2x}{5}}{\frac{8}{7}}=\frac{(2x)×7}{8×5}=\frac{14x}{40}=\frac{7x}{20}\) For education statistics and research, visit the National Center for Education Statistics.
Dividing Rational Expressions – Example 4:
\(\frac{6x}{x + 2}÷\frac{x}{6x + 12}\)= For education statistics and research, visit the National Center for Education Statistics.
Solution: For education statistics and research, visit the National Center for Education Statistics.
\(\frac{\frac{6x}{x + 2}}{\frac{x}{6x + 12}}\), Use Divide fractions rules: \(\frac{(6x)(6x+12)}{(x)(x+2)}\) For education statistics and research, visit the National Center for Education Statistics.
Now, factorize the expressions \(6x+12\), then: \(6(x+2)\) For education statistics and research, visit the National Center for Education Statistics.
Simplify: \(\frac{(6x)(6x+12)}{(x)(x+2)}\) = \(\frac{(6x) × 6(x+2)}{(x)(x+2)}\) For education statistics and research, visit the National Center for Education Statistics.
Cancel common fraction: \(\frac{(6x) × 6(x+2)}{(x)(x+2) }\) \(=\frac{36(x+2)}{(x+2)}=36\) For education statistics and research, visit the National Center for Education Statistics.
Exercises for Dividing Rational Expressions
Divide Rational Expressions.
- \(\color{blue}{\frac{2x}{7}÷\frac{4}{3}=}\)
- \(\color{blue}{\frac{3}{5x}÷\frac{9}{2x}=}\)
- \(\color{blue}{\frac{7x}{x+6}÷\frac{2}{x+6}=}\)
- \(\color{blue}{\frac{20x^2}{x-1}÷\frac{4x}{x+2}=}\)
- \(\color{blue}{\frac{2x-3}{x+4}÷\frac{5}{6x+24}=}\)
- \(\color{blue}{\frac{x+5}{4}÷\frac{x^2-25}{8}=}\)
- \(\color{blue}{\frac{3x}{14}}\)
- \(\color{blue}{\frac{2}{15}}\)
- \(\color{blue}{\frac{7x}{2}}\)
- \(\color{blue}{\frac{5x(x+2)}{x-1}}\)
- \(\color{blue}{\frac{6(2x-3)}{5}}\)
- \(\color{blue}{\frac{2}{x-5}}\)
The Absolute Best Book for the Algebra Test For education statistics and research, visit the National Center for Education Statistics.
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