ACT Math: Factor Any Polynomial Using the Reverse FOIL Method

Factoring quadratics like x^2+5x+6 is just reversing FOIL. (1) Write the quadratic in standard form ax^2+bx+c. (2) Find two numbers that multiply to give c and add to give b. (3) Rewrite the middle term using those two numbers. (4) Group the first two terms and the last two terms. (5) Factor out the greatest common factor from each group. This method works for every factorable quadratic because it reduces factoring to a searchable arithmetic problem.

Example: x^2+7x+12. Step 1: Already in standard form. Step 2: Find two numbers that multiply to 12 and add to 7. Check: 3 and 4 (3×4=12, 3+4=7). Step 3: Rewrite as x^2+3x+4x+12. Step 4: Group as (x^2+3x)+(4x+12). Step 5: Factor each group: x(x+3)+4(x+3)=(x+4)(x+3). Check by FOIL: (x+4)(x+3)=x^2+3x+4x+12=x^2+7x+12. Correct.

Shortcut 1: Difference of squares. x^2-9=(x-3)(x+3). If you see a^2-b^2, the factors are (a-b)(a+b). Shortcut 2: Perfect square trinomials. x^2+6x+9=(x+3)^2 and x^2-6x+9=(x-3)^2. If the first and last terms are perfect squares and the middle term is 2 times their product, use this. Shortcut 3: Factor out the GCF first. 3x^2+6x=3x(x+2). Always check for a common factor before factoring. Shortcut 4: If a≠1, use grouping or the AC method. For 2x^2+7x+3, multiply a and c: 2×3=6. Find two numbers that multiply to 6 and add to 7: 6 and 1. Rewrite as 2x^2+6x+x+3, then group. Knowing these four shortcuts saves 30 seconds per problem on test day.

Practice recognizing these patterns: Is it a^2-b^2? Is it (x+a)^2? Is there a GCF? These three questions cover 70% of factoring questions on ACT Math.

Problem 1: x^2+8x+15. Problem 2: x^2-16. Problem 3: x^2+10x+25. Problem 4: 2x^2+9x+4. Problem 5: x^2-5x+6. Problem 6: 3x^2+12x. Problem 7: x^2+2x-8. Problem 8: 4x^2-1. Problem 9: x^2-11x+30. Problem 10: 5x^2+20x+15. For each, factor completely and check your answer by FOIL (or distributing). If you miss any, redo it and identify which step broke. Do this drill twice this week; speed improves dramatically with repetition.

Answers: 1) (x+3)(x+5). 2) (x-4)(x+4). 3) (x+5)^2. 4) (2x+1)(x+4). 5) (x-2)(x-3). 6) 3x(x+4). 7) (x+4)(x-2). 8) (2x-1)(2x+1). 9) (x-5)(x-6). 10) 5(x^2+4x+3)=5(x+1)(x+3).

Factoring appears on every ACT Math section because it is a prerequisite for solving equations, simplifying rational expressions, and finding roots. Many harder problems hinge on your ability to factor quickly. One hour spent mastering these five steps will unlock 3-5 points across the whole test because factoring is foundational.

Commit to the ten-problem drill this week. By test day, factoring will feel so automatic that you solve (x-2)(x-3)=0 instantly without thinking.