Overview
Rational expressions are fractions whose numerators and/or denominators are polynomials. The SAT tests simplifying these expressions, performing arithmetic operations (adding, subtracting, multiplying, dividing), and solving rational equations. A consistent theme is identifying values that make a denominator zero — these are excluded from the domain and can generate extraneous solutions when solving. Questions at the hard level frequently include a trap where an algebraic solution is extraneous, and that trap answer appears as an answer choice.
Key Points
1. Simplifying Rational Expressions
Method: Factor completely, then cancel common factors.
Step 1: Factor the numerator completely.
Step 2: Factor the denominator completely.
Step 3: Cancel factors that appear in both numerator and denominator.
Step 4: Note any domain restrictions before canceling.
Example:
(x² - 9) / (x² - x - 6)
= (x+3)(x-3) / [(x-3)(x+2)]
= (x+3) / (x+2), x ≠ 3, x ≠ -2
2. Multiplying and Dividing Rational Expressions
Multiply: (A/B) · (C/D) = AC / BD → factor and cancel before multiplying
Divide: (A/B) ÷ (C/D) = (A/B) · (D/C) = AD / BC
Strategy: Factor all expressions first, flip the second fraction for division, then cancel before multiplying.
3. Adding and Subtracting Rational Expressions
Step 1: Find the LCD (Least Common Denominator).
Step 2: Rewrite each fraction with the LCD as its denominator.
Step 3: Add or subtract numerators; keep the LCD.
Step 4: Simplify the resulting expression.
Example:
1/(x-2) + 3/(x+1)
LCD = (x-2)(x+1)
= (x+1)/[(x-2)(x+1)] + 3(x-2)/[(x-2)(x+1)]
= [(x+1) + 3(x-2)] / [(x-2)(x+1)]
= (4x - 5) / [(x-2)(x+1)]
4. Solving Rational Equations (5-Step Method)
Step 1: Identify excluded values — set each denominator = 0 and note those x-values.
Step 2: Find the LCD of all fractions in the equation.
Step 3: Multiply every term on both sides by the LCD to clear fractions.
Step 4: Solve the resulting polynomial equation.
Step 5: Check every solution against the excluded values (and substitute back).
Example:
x/(x-3) = 12/(x-3) + 2
Excluded value: x ≠ 3
LCD = (x-3)
Multiply through: x = 12 + 2(x-3)
x = 12 + 2x - 6
x = 6 + 2x
-x = 6
x = -6 ← not excluded, so valid. ✓
5. Domain and Undefined Values
A rational expression is undefined wherever any denominator equals zero. Those x-values are excluded from the domain.
| Expression | Excluded values |
|---|---|
1/(x-5) | x = 5 |
x/[(x+2)(x-1)] | x = -2, x = 1 |
(x²-4)/(x-2) | x = 2 (even though numerator also has factor) |
Pitfalls and Common Mistakes
Pitfall 1: Accepting Extraneous Solutions
Description: After clearing the LCD and solving, the algebraic answer makes a denominator zero in the original equation — it is extraneous.
Example: Solving 3/(x-4) = 1 gives x-4 = 3, so x = 7. But if a different equation gives x = 4 and the original has (x-4) in the denominator, x = 4 is extraneous (division by zero).
Fix: Always check every solution by substituting into the original equation. If any denominator becomes zero, discard that solution. On the SAT, extraneous solutions are frequently placed as answer choices to trap students.
Pitfall 2: Failing to Distribute the Negative When Subtracting Fractions
Description: When subtracting a rational expression with more than one term in the numerator, students forget to distribute the minus sign across all terms.
Example:
x/(x+1) - (x-3)/(x+1) ≠ (x - x - 3)/(x+1)
The correct answer is (x - (x-3))/(x+1) = 3/(x+1), not (2x-3)/(x+1).
Fix: Put parentheses around the entire numerator being subtracted, then distribute the minus sign: x - (x-3) = x - x + 3 = 3.
Pitfall 3: Canceling Terms Instead of Factors
Description: Students cancel individual terms across addition/subtraction rather than canceling common polynomial factors.
Example: Writing (x+3)/3 = x by “canceling the 3s” — this is wrong.
Fix: You can only cancel a common factor that multiplies the entire numerator and the entire denominator. (x+3)/3 has no common factor to cancel because 3 does not factor out of x+3 as a whole.
Pitfall 4: Ignoring Domain Restrictions After Simplifying
Description: After canceling a common factor, students forget that the original domain restriction still applies.
Example: (x²-9)/(x-3) = x+3 — but x = 3 is still excluded even though the simplified form appears defined there.
Fix: Note all excluded values from the original expression before simplifying. Those restrictions carry through to the simplified form.
Related Entries
- Radicals_Rational_Exponents — Negative exponents produce rational expressions; the same “undefined” domain concerns apply.
- Quadratic_Equations — Factoring quadratics is the primary tool for simplifying rational expressions.
- Polynomial_Operations_Factoring — Long division or synthetic division applies when the numerator degree exceeds the denominator degree.
- Systems_Nonlinear_Equations — Rational equations with two variables may appear as systems requiring substitution.
- Functions_Concepts_Notation — The domain of a function defined by a rational expression excludes all values where the denominator is zero.
Quick Reference Card
| Operation | Procedure |
|---|---|
| Simplify | Factor fully → cancel common factors |
| Multiply | Multiply numerators, multiply denominators → simplify |
| Divide | Flip second fraction → multiply → simplify |
| Add / Subtract | Find LCD → convert → combine numerators → simplify |
| Solve equation | Note excluded values → multiply by LCD → solve → check |
| Extraneous solution | Makes a denominator zero in original equation → discard |
| Undefined | Expression has no value where any denominator = 0 |
| Domain | All real numbers except values making any denominator = 0 |
| SAT trap | Extraneous solution is typically offered as an answer choice |