ACT Math: Master Rational Expressions with the Common Denominator Method

A rational expression is a fraction with polynomials in the numerator and denominator. To add, subtract, or simplify, find the common denominator (just like adding fractions 1/2+1/3=3/6+2/6=5/6). Step 1: factor the numerator and denominator completely. Step 2: identify the least common denominator (LCD), rewrite each fraction using LCD, combine, and simplify. This method is identical to fraction addition; you're simply working with algebraic fractions instead of numbers.

Example: Simplify (x+2)/(x^2-4)+(1)/(x-2). Factor: x^2-4=(x+2)(x-2). The LCD is (x+2)(x-2). Rewrite: (x+2)/((x+2)(x-2))+(1)/(x-2)=(x+2)/((x+2)(x-2))+(x+2)/((x+2)(x-2))=(x+2+x+2)/((x+2)(x-2))=(2x+4)/((x+2)(x-2)). Factor the numerator: 2x+4=2(x+2). Simplify: (2(x+2))/((x+2)(x-2))=2/(x-2).

Mistake 1: Canceling terms instead of factors. You can only cancel factors that appear in both numerator and denominator completely. You cannot cancel (x+2) from (x+2+y)/(x+2-z). Mistake 2: Forgetting to factor before simplifying. Many students see (x^2-4)/(x-2) and don't factor x^2-4=(x+2)(x-2), missing the cancellation. Mistake 3: Making arithmetic errors when finding LCD. Write out each denominator's factors and list each unique factor once. Always factor first, identify what cancels, then simplify. Never skip the factoring step.

On test day, force yourself to write out every factoring step, even if it feels slow. Factoring takes 20 seconds but prevents errors worth 1-2 points.

Problem 1: (x^2-9)/(x-3). Factor: (x+3)(x-3)/(x-3). Cancel: x+3. Problem 2: (2x)/(x^2+x)+(1)/(x). Factor denominator: x(x+1). LCD: x(x+1). Rewrite: (2x)/(x(x+1))+(x+1)/(x(x+1))=(2x+x+1)/(x(x+1))=(3x+1)/(x(x+1)). Problem 3: (x^2-1)/(x^2+2x+1). Factor: (x+1)(x-1)/((x+1)^2). Cancel one (x+1): (x-1)/(x+1). For each problem, show all factoring and cancellation steps before writing your final answer.

Speed comes after accuracy. Master these five, then try five more until simplification feels automatic.

Rational expression questions appear on most ACT Math sections. Many students avoid them or guess because the algebra looks intimidating. With the common denominator method and factoring as your foundation, rational expressions become routine arithmetic with letters instead of numbers.

Spend this week practicing factoring and LCD identification. By test day, you will confidently simplify any rational expression in 2-3 minutes.