Curriculum fit: US Algebra 2. Aligned to Common Core HSA-APR.D.6 (rewrite rational expressions) and HSA-REI.A.2 (solve rational equations and check for extraneous solutions). Check your own state standards and course.
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What Is It?
A rational expression is a fraction with polynomials on the top and bottom, such as (x + 3) ÷ (x + 4). Because you can never divide by zero, some values of x are not allowed.
To simplify, factor the top and bottom and cancel common factors. To add or subtract, find a common denominator first. To solve an equation, multiply both sides by the common denominator, solve, then check each answer.
Checking matters: a solution that makes a denominator zero is extraneous and must be thrown out.
Key Formulas and Ideas
- Simplify: factor, then cancel common factors (not common terms).
- Add: a/b + c/d = (ad + bc) ÷ bd, or use the lowest common denominator (LCD).
- Multiply: multiply the tops and the bottoms, after factoring.
- Divide: multiply by the reciprocal of the second fraction.
- Domain: exclude any x that makes a denominator equal to 0.
Worked Examples
Simplify and State the Restrictions
Simplify (x² − 9) ÷ (x² + x − 12).
- Factor: (x − 3)(x + 3) ÷ [(x + 4)(x − 3)].
- Cancel the common factor x − 3, leaving (x + 3) ÷ (x + 4).
- The original denominator is zero when x = 3 or x = −4, so these are excluded.
✓ Answer (x + 3) ÷ (x + 4), with x ≠ 3 and x ≠ −4
Add Two Rational Expressions
Simplify 1/x + 1/(x + 2).
- The LCD is x(x + 2).
- Rewrite: (x + 2) ÷ [x(x + 2)] + x ÷ [x(x + 2)].
- Add the tops: (2x + 2) ÷ [x(x + 2)].
✓ Answer (2x + 2) ÷ [x(x + 2)]
Solve and Check for Extraneous Solutions
Solve x² ÷ (x − 3) = 9 ÷ (x − 3).
- Multiply both sides by x − 3 (x ≠ 3): x² = 9.
- So x = 3 or x = −3.
- x = 3 makes the denominator zero, so reject it.
✓ Answer x = −3
Common Mistakes to Avoid
- Cancelling terms instead of factors. In (x + 3) ÷ (x + 4) you cannot cancel the x.
- Forgetting to state the values that x cannot take.
- Not checking answers, and keeping an extraneous solution.
- Adding fractions by adding the denominators.
Practice Questions
Frequently Asked Questions
What is an extraneous solution?
It is an answer that comes out of your algebra but does not work in the original equation, usually because it makes a denominator zero. Always substitute back to check.
Can I cancel the x in (x + 3) ÷ (x + 4)?
No. You can only cancel factors that multiply the whole top and the whole bottom. The x here is part of a sum, so it cannot be cancelled.
How do I find the LCD?
Factor each denominator, then take every factor the greatest number of times it appears in any denominator.
Keep Going With These Guides
Algebra 1: Factoring Polynomials Step by Step
How to factor in Algebra 1: common factors, trinomials, grouping, difference of squares and perfect squares, with checked worked examples.
Middle SchoolHigh SchoolSolving Quadratic Equations: Factorising and the Formula
Solve quadratics by factorising and with the quadratic formula, using worked examples. Covers GCSE quadratic equations and Algebra 1 and 2 methods.
GCSEA-LevelHigh SchoolSimplifying Algebraic Expressions: Collecting Like Terms
Collect like terms, multiply and divide algebraic terms and simplify expressions with brackets, with step-by-step examples and practice questions.
KS3GCSEMiddle SchoolHigh SchoolAlgebra 2: Polynomial Division and the Remainder Theorem
Synthetic division, the remainder theorem and the factor theorem in Algebra 2, with worked examples to find roots and factors.
High SchoolSources and Review
- Common Core: High School Algebra, Arithmetic with Polynomials
- Common Core: High School Algebra, Reasoning with Equations and Inequalities
Reviewed by StudyMorrow editorial review on 6 Nov 2026.