ACT Math Rational Expressions: Simplify and Solve Without Mistakes

Step 1: Factor the numerator and denominator completely. Example: (x^2+5x+6)/(x^2-4)=((x+2)(x+3))/((x+2)(x-2)). Step 2: Cancel common factors. The (x+2) cancels: (x+3)/(x-2). Step 3: Identify restrictions (values of x that make the denominator zero). Here, x≠2 and x≠-2 (original denominator). This three-step method handles every rational expression on the ACT because it's mechanical: factor, cancel, restrict. Most students skip factoring and jump to canceling, causing errors. Always factor first.

Another example: (2x^2-8)/(2x+4)=2(x^2-4)/2(x+2)=2(x+2)(x-2)/2(x+2)=(x-2). Factor first, cancel the common 2 and (x+2), leaving (x-2). This careful approach prevents errors.

Mistake 1: Canceling before factoring. If you see (x^2+5x+6)/(x+2) and try to cancel x+2 immediately, you'll make errors. Factor first: ((x+2)(x+3))/(x+2), then cancel to get (x+3). Mistake 2: Forgetting to identify restrictions. If the denominator had (x-2), you must note x≠2. Some questions ask for restrictions specifically. Mistake 3: Canceling terms that aren't factors. You can't cancel x from x+2; x is not a factor of (x+2). Only common factors can cancel. Factor completely and verify factors before canceling anything.

Create a reference showing factoring patterns (difference of squares, trinomials, GCF). Reference it daily this week. By test day, factoring will be automatic enough that you factor before you even think about canceling.

Problem 1: (x^2-4)/(x-2). Factor: ((x+2)(x-2))/(x-2). Cancel: x+2 (x≠2). Problem 2: (x^2+5x+6)/(x^2-4). Factor: ((x+2)(x+3))/((x+2)(x-2)). Cancel: (x+3)/(x-2) (x≠-2, 2). Problem 3: (2x^2-8)/(x+2). Factor: 2(x^2-4)/(x+2)=2(x+2)(x-2)/(x+2). Cancel: 2(x-2) (x≠-2). Problem 4: (x^2-9)/(x^2-6x+9). Factor: ((x+3)(x-3))/((x-3)^2). Cancel: (x+3)/(x-3) (x≠3). Problem 5: (3x^2+6x)/(x+2). Factor: 3x(x+2)/(x+2). Cancel: 3x (x≠-2). Simplify all five, showing each factoring step and identifying restrictions.

Find five rational expression questions from a practice test. For each, factor completely, cancel common factors, and identify restrictions. By the fifth question, the three-step method will feel automatic.

Rational expressions appear on most ACT Math tests, usually in questions 35-50. Once you master the three-step method, these problems are mechanical. Students who simplify rational expressions systematically pick up 1-2 points because the method is straightforward once you commit to factoring first.

Drill rational expressions daily this week. Each day, simplify five expressions using the three-step method. By test day, you should simplify any rational expression correctly within 60 seconds.