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:
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
- Geometry Puzzle – Challenge 65
- The Ultimate SAT Math Formula Cheat Sheet
- The Ultimate SIFT Math Course (+FREE Worksheets & Tests)
- The Ultimate Adults Math Refresher Course (+FREE Worksheets & Tests)
- How to Master the Pythagorean Theorem and Right Triangles
- The Quotient Rule: Not Just Dividing Derivatives But Simple Enough
- What to Consider when Retaking the ACT or SAT?
- The Best Calculators for Grade 9 Students
- Intelligent Math Puzzle – Challenge 85
- Full-Length 7th Grade GMAS Math Practice Test-Answers and Explanations
What people say about "How to Divide Rational Expressions? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.