Rational Expressions — SAT Math Guide

Q: What You Need to Know

• A rational expression is a fraction with polynomials in the numerator and denominator • Factor polynomials completely before simplifying rational expressions • Multiply rational expressions by multiplying numerators together and denominators together • Divide rational expressions by multiplying by

Rational expressions SAT questions appear regularly in the Advanced Math domain of the Digital SAT. These problems involve fractions where both the numerator and denominator contain polynomials. You'll typically see 2-3 rational expression questions on your SAT math section, making them a valuable topic to master. With the right approach, these problems become much more manageable than they first appear.

What You Need to Know

A rational expression is a fraction with polynomials in the numerator and denominator

Factor polynomials completely before simplifying rational expressions

Multiply rational expressions by multiplying numerators together and denominators together

Divide rational expressions by multiplying by the reciprocal of the divisor

Add and subtract rational expressions by finding a common denominator

Always check for restrictions where the denominator equals zero

📐 KEY FORMULA: (a/b) ÷ (c/d) = (a/b) × (d/c) = (ad)/(bc)

💡 PRO TIP: Factor first, then cancel common factors — this prevents messy arithmetic and reduces errors on the Digital SAT.

How to Solve Rational Expressions on the SAT

Example Question 1 — Medium Difficulty

Which of the following is equivalent to (x² - 4)/(x + 2) · (x + 3)/(x - 2)?

A) (x + 3)

B) (x - 3)

C) (x + 2)(x + 3)

D) (x + 3)/(x - 2)

Solution:

Step 1: Factor x² - 4 as (x + 2)(x - 2)

Step 2: Rewrite as [(x + 2)(x - 2)]/(x + 2) · (x + 3)/(x - 2)

Step 3: Cancel (x + 2) from numerator and denominator, then cancel (x - 2)

✅Answer: A — After canceling common factors, only (x + 3) remains.

Example Question 2 — Hard Difficulty

If f(x) = (2x² - 8)/(x² - 5x + 6) and g(x) = (x - 3)/(x + 2), what is f(x)/g(x) for x ≠ 2, 3, -2?

A) 2

B) 2(x + 2)

C) 2(x - 2)

D) (x + 2)/(x - 3)

Solution:

Step 1: Factor f(x): numerator = 2(x² - 4) = 2(x + 2)(x - 2), denominator = (x - 2)(x - 3)

Step 2: So f(x) = [2(x + 2)(x - 2)]/[(x - 2)(x - 3)] = [2(x + 2)]/(x - 3)

Step 3: Calculate f(x)/g(x) = [2(x + 2)]/(x - 3) ÷ (x - 3)/(x + 2) = [2(x + 2)]/(x - 3) · (x + 2)/(x - 3)

Step 4: Simplify: [2(x + 2)²]/[(x - 3)²] · [(x - 3)]/[(x + 2)] = 2(x + 2)

✅Answer: B — After dividing and simplifying, we get 2(x + 2).

Common SAT Math Mistakes to Avoid

❌Mistake: Canceling terms instead of factors (like canceling x from x + 2 and x + 3)

✅Fix: Only cancel common factors that multiply the entire numerator or denominator

❌Mistake: Forgetting to factor completely before simplifying

✅Fix: Always factor polynomials fully, looking for difference of squares, perfect square trinomials, and common factors

❌Mistake: Adding fractions by adding numerators and denominators separately

✅Fix: Find the least common denominator first, then add numerators only

❌Mistake: Ignoring domain restrictions when denominators equal zero

✅Fix: Note which values make denominators zero — these are excluded from the domain

Practice Question — Try It Yourself

What is (x² + 5x + 6)/(x + 2) - (x + 1) equal to?

A) 2

B) x + 1

C) x + 2

D) x + 3

Show Answer

Answer: A — Factor the numerator as (x + 2)(x + 3), cancel (x + 2), get (x + 3) - (x + 1) = 2

Key Takeaways for the SAT

Factor polynomials completely before attempting any operations with rational expressions

SAT math rational expressions problems often have elegant solutions after proper factoring

When multiplying or dividing, look for opportunities to cancel common factors early

Remember that rational expressions follow the same rules as numeric fractions

College Board often tests whether you can recognize equivalent forms of the same expression