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Rational Expressions for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about rational expressions in one place: simplifying, multiplying, dividing, adding, and subtracting rational expressions, excluded values, and solving rational equations — with step-by-step examples, worked problems, and a free practice quiz.
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Free · No signupRational Expressions
Simplifying, operations, excluded values, and rational equations.
What is a rational expression?
A rational expression is a fraction where the numerator, denominator, or both contain variables, like (x + 3) / (x² − 9). Rational expressions follow the same rules as numerical fractions — simplify by canceling common factors, and never divide by zero.
Factoring is the key skill behind almost every rational expression question — before doing anything else, factor the numerator and denominator fully.
Simplifying rational expressions
Simplify: (x² − 9) / (x² + x − 6)
Only fully factored common factors can be canceled — never cancel individual terms that are added or subtracted inside an unfactored expression. (x² + 3) / x cannot be simplified by canceling the x's.
Multiplying & dividing
Multiply: (x/4) · (8 / (x+2))
Just like with numerical fractions, dividing by a rational expression means multiplying by its reciprocal — flip the second expression, then multiply as usual.
Adding & subtracting
Just like numerical fractions, rational expressions need a common denominator before adding or subtracting.
Add: 3/x + 2/(x+1)
Only the numerators combine when adding or subtracting fractions with a common denominator — the denominator itself is never added or changed once it's shared.
Excluded values
Any value of x that makes a denominator equal zero must be excluded from the domain, since division by zero is undefined.
What values of x are excluded from (x + 1) / (x² − 5x + 6)?
Find excluded values from the original, unsimplified denominator. If a factor cancels during simplification, that value is still excluded, even though it may no longer appear in the simplified expression.
Solving rational equations
Solve: 4/x + 1/2 = 3/x
After solving a rational equation, always check the solution against the excluded values from the original denominators. If a solution matches an excluded value, it must be rejected as extraneous.
Common PSAT traps
- Canceling terms instead of factors: only fully factored common expressions can be canceled, never individual terms inside a sum.
- Finding excluded values from the simplified expression: always check the original denominator before any cancellation.
- Forgetting to check for extraneous solutions: a solution to a rational equation that makes any original denominator zero must be rejected.
- Adding denominators when combining fractions: only numerators combine over a shared denominator.
Test day strategy for rational expressions
| Question signal | Fastest approach |
|---|---|
| Simplify a rational expression | Factor numerator and denominator fully, then cancel common factors |
| Multiply or divide rational expressions | Factor first, then multiply straight across (or flip and multiply for division) |
| Add or subtract rational expressions | Find a common denominator before combining numerators |
| "For what value of x is this undefined" | Set the original (unfactored) denominator equal to zero |
| Rational equation to solve | Clear denominators first, solve, then check for extraneous solutions |
Now put it to work
One comprehensive quiz covering the full topic — simplifying, operations, excluded values, and rational equations.
Rational Expressions
Simplifying, operations, excluded values, and rational equations.