PSAT/NMSQT Math: Rational Expressions
A rational expression is a fraction with polynomials in the numerator, denominator, or both, and simplifying one safely requires factoring first, canceling correctly, and always keeping track of which values would make the denominator zero. This free, complete lesson covers simplifying rational expressions, factoring before canceling, domain restrictions, and evaluating rational functions. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Simplifying Rational Expressions
To simplify a rational expression, cancel any factor that is common to both the numerator and the denominator. This is only valid for factors that multiply the entire numerator and denominator, never for terms that are only added or subtracted.
Worked Example: Simplify 6x/(3x).
6x/3x = 2 (canceling the common factor 3x, valid since x ≠ 0)
1. Simplify: 6x/(3x)
Explanation: Both terms share a factor of 3x: 6x/3x = 2.
2. Simplify: (2x² + 4x)/(4x)
Explanation: Factor out 2x from the numerator: 2x(x + 2)/4x. Cancel the common factor 2x: (x + 2)/2.
3. Simplify: x⁻²/x⁻⁵
Explanation: Subtracting exponents: −2 − (−5) = 3, so x⁻²/x⁻⁵ = x³.
2. Factoring Before Canceling
When both the numerator and denominator are more complex polynomials, factor both completely first, then cancel any matching factors. Canceling individual terms without factoring (such as canceling an x that appears in a sum, not as a full factor) produces an incorrect result.
Worked Example: Simplify (x² − 9)/(x² − 3x).
Factor the numerator (difference of squares) and denominator (common factor):
(x + 3)(x − 3) / x(x − 3) = (x + 3)/x (canceling x − 3, valid since x ≠ 3)
1. Simplify: (x² − 9)/(x² − 3x) (see worked example above)
Explanation: Factoring both: (x + 3)(x − 3) / x(x − 3). Canceling the common factor (x − 3) leaves (x + 3)/x.
2. Which expression cannot be simplified by canceling common factors?
Explanation: In (x + 5)/(x + 2), the numerator and denominator share no common factor since neither (x + 5) nor (x + 2) can be factored further, and they don't match; the x's here are only terms in a sum, not a shared multiplied factor.
3. Simplify: (x² + 5x + 6)/(x² − 4)
Explanation: Factor both: (x + 2)(x + 3) / (x + 2)(x − 2). Canceling the common factor (x + 2) leaves (x + 3)/(x − 2).
3. Domain Restrictions
A rational expression is undefined wherever its denominator equals zero, since division by zero is never allowed. Always find these restricted values before or after simplifying, they remain excluded from the domain even if a factor causing them gets canceled out algebraically.
Worked Example: What values of x make 5/(x² − 16) undefined?
Set the denominator equal to 0: x² − 16 = 0 → x² = 16.
x = 4 or x = −4
1. What values of x make 5/(x² − 16) undefined? (see worked example above)
Explanation: Setting the denominator to 0: x² = 16 → x = ±4.
2. What is the domain restriction for E(v) = 300/(v + 5)?
Explanation: The denominator v + 5 equals 0 when v = −5, so v cannot equal −5.
3. Before canceling, (x² − 9)/(x² − 3x) has denominator x(x − 3). Which values are excluded from its domain?
Explanation: The original (unsimplified) denominator x(x − 3) equals 0 when x = 0 or x = 3, so both values are excluded, even though the simplified expression (x + 3)/x no longer visibly shows the x = 3 restriction.
4. Evaluating Rational Functions and Proportions
Evaluating a rational function works exactly like evaluating any other function, substitute the given value and simplify. A proportion, a/b = c/d, can be rewritten by cross-multiplying: ad = bc.
Worked Example: If f(x) = (x² − 1)/(x + 1), find f(3).
f(3) = (9 − 1)/(3 + 1) = 8/4 = 2
1. If f(x) = (x² − 1)/(x + 1), find f(3). (see worked example above)
Explanation: f(3) = (9 − 1)/(3 + 1) = 8/4 = 2.
2. If a/b = c/d, which of the following is equivalent?
Explanation: Cross-multiplying a proportion always gives the product of the outer terms equal to the product of the inner terms: ad = bc.
3. Simplify the complex fraction 1/(x²/2).
Explanation: Dividing by a fraction means multiplying by its reciprocal: 1 · (2/x²) = 2/x².
Common Mistakes to Avoid
- Canceling a term instead of a factor. You can only cancel something that multiplies the entire numerator and denominator, never a term that's only added or subtracted within a sum.
- Simplifying before factoring completely. Always factor both the numerator and denominator fully first, so no common factor is missed.
- Forgetting that a domain restriction still applies after simplifying. A value excluded from the original (unsimplified) denominator stays excluded, even if the simplified expression no longer visibly shows it.
- Cross-multiplying incorrectly in a proportion, mismatching which numerator goes with which denominator.
- Forgetting to flip the second fraction when dividing rational expressions. Dividing by a fraction always means multiplying by its reciprocal.
Frequently Asked Questions
How do I know when I'm allowed to cancel something in a rational expression?
Only cancel a factor that multiplies the entire numerator and the entire denominator. If the term is part of a sum or difference rather than a multiplied factor, it cannot be canceled directly, factor first instead.
Why do domain restrictions matter if they disappear after simplifying?
The simplified expression is only equivalent to the original for the values where the original was actually defined. The excluded value is still a "hole" in the function, even though it's no longer visible in the simplified form.
Where can I practice more problems like these?
The Rational Expressions quiz in the PSAT/NMSQT Math Question Bank includes additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.