Divide Rational Expressions Calculator
Divide rational expressions using Keep-Change-Flip with complete step-by-step solutions
Dividend (First Expression)
Divisor (Second Expression)
How to Divide Rational Expressions
The Keep-Change-Flip Rule
Division of rational expressions uses the same rule as dividing numerical fractions:
- Keep the first fraction as is
- Change division to multiplication
- Flip the second fraction (multiply by its reciprocal)
Complete Process
- Apply Keep-Change-Flip to convert division to multiplication
- Factor all numerators and denominators
- Cross-cancel common factors
- Multiply remaining factors
- State all restrictions (including from the divisor's numerator!)
Important Note About Restrictions
When dividing, remember three sources of restrictions:
- First fraction's denominator ≠ 0
- Second fraction's denominator ≠ 0
- Second fraction's numerator ≠ 0 (you're dividing by this!)
Example
Divide: (x² - 1)/(x + 2) ÷ (x + 1)/(x² - 4)
- Keep-Change-Flip: (x² - 1)/(x + 2) × (x² - 4)/(x + 1)
- Factor: [(x+1)(x-1)]/(x+2) × [(x+2)(x-2)]/(x+1)
- Cancel (x+1) and (x+2)
- Result: (x - 1)(x - 2), where x ≠ -2, -1, ±2
Frequently Asked Questions
Why do we flip the second fraction?
Division by a fraction is the same as multiplication by its reciprocal. This is a fundamental property of fractions: (a/b) ÷ (c/d) = (a/b) × (d/c).
What's the difference between dividing and multiplying rational expressions?
The only difference is the first step. For division, you must apply Keep-Change-Flip first. After that, both operations follow the same process: factor, cancel, multiply.
Why is the divisor's numerator a restriction?
Because you're dividing by the entire second fraction. If its numerator is zero, you're dividing by zero, which is undefined.
Can I skip the Keep-Change-Flip step?
No! You must convert division to multiplication first. There's no direct way to divide rational expressions without this step.
Do I flip before or after factoring?
Always flip first, then factor. The Keep-Change-Flip rule must be applied to the original expressions.
What if the divisor is just a polynomial (not a fraction)?
Treat it as a fraction with denominator 1. For example, dividing by (x + 2) is the same as dividing by (x + 2)/1, which flips to 1/(x + 2).
Can the result of division be larger than the dividend?
Yes! Unlike numerical fractions (where dividing makes things smaller), rational expressions can result in larger-degree polynomials.
How do I check my answer?
Multiply your answer by the divisor. If you get the dividend back (considering restrictions), your answer is correct.