How to Divide Rational Expressions? (+FREE Worksheet!)

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: \(\frac{5x}{x + 3}÷\frac{x}{2x + 6}=\frac{5x}{x + 3}×\frac{2x + 6}{x}=\frac{5x(2x + 6)}{x(x+3)}\)
Now, factorize the expressions \(2x+6\), then: \(2(x+3)\)
Simplify: \(\frac{5x(2x + 6)}{x(x+3)}\) =\(\frac{5x×2(x+3)}{x(x+3)}\)
Cancel common factor: \(\frac{5x×2(x+3)}{x(x+3)}=\frac{10x(x+3)}{x(x+3)}=10\)
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: \(\frac{(6x)(6x+12)}{(x)(x+2)}\)
Now, factorize the expressions \(6x+12\), then: \(6(x+2)\)
Simplify: \(\frac{(6x)(6x+12)}{(x)(x+2)}\) = \(\frac{(6x) × 6(x+2)}{(x)(x+2)}\)
Cancel common fraction: \(\frac{(6x) × 6(x+2)}{(x)(x+2) }\) \(=\frac{36(x+2)}{(x+2)}=36\)
Exercises for Dividing Rational Expressions
Divide Rational Expressions.
- \(\color{blue}{\frac{2x}{7}÷\frac{4}{3}=}\)
- \(\color{blue}{\frac{3}{5x}÷\frac{9}{2x}=}\)
- \(\color{blue}{\frac{7x}{x+6}÷\frac{2}{x+6}=}\)
- \(\color{blue}{\frac{20x^2}{x-1}÷\frac{4x}{x+2}=}\)
- \(\color{blue}{\frac{2x-3}{x+4}÷\frac{5}{6x+24}=}\)
- \(\color{blue}{\frac{x+5}{4}÷\frac{x^2-25}{8}=}\)

- \(\color{blue}{\frac{3x}{14}}\)
- \(\color{blue}{\frac{2}{15}}\)
- \(\color{blue}{\frac{7x}{2}}\)
- \(\color{blue}{\frac{5x(x+2)}{x-1}}\)
- \(\color{blue}{\frac{6(2x-3)}{5}}\)
- \(\color{blue}{\frac{2}{x-5}}\)
The Absolute Best Book for the Algebra Test
Related to This Article
More math articles
- The Ultimate ATI TEAS 7 Math Formula Cheat Sheet
- Overview of the TSI Mathematics Test
- 6th Grade MCA Math Worksheets: FREE & Printable
- Calculations to Help Curb Your Rising Debt
- 7th Grade New York State Assessments Math Worksheets: FREE & Printable
- How to Find the Surface Area of Pyramid?
- PSAT 8/9 Math Worksheets: FREE & Printable
- How to Use Input/output Tables to Add and Subtract Integers?
- 4th Grade Georgia Milestones Assessment System Math FREE Sample Practice Questions
- Interwoven Variables: The World of Implicit Relations
What people say about "How to Divide Rational Expressions? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.