Factoring Trinomials: Step-by-Step Guide
A trinomial like ax^2 + bx + c has three terms. Many trinomials can be written as a product of two binomials, like (x + 2)(x + 3). When the number in front of x^2 is 1, find two numbers that multiply to c and add to b. When it is not 1, use the AC method: find two numbers that multiply to a x c and add to b, split the middle term, and factor by grouping.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Pull out any common factor first. See if all three terms share a number or letter. If they do, factor it out before you do anything else.\[ax^{2}+bx+c = g\left(\tfrac{a}{g}x^{2}+\tfrac{b}{g}x+\tfrac{c}{g}\right)\]
- Find the two special numbers. If a = 1, find two numbers that multiply to c and add to b. If a is not 1, find two numbers that multiply to a x c and add to b.\[p\cdot q = ac,\quad p+q = b\]
- Split the middle term. Rewrite the middle term bx as two terms using your two numbers. Now you have four terms.\[ax^{2}+bx+c = ax^{2}+px+qx+c\]
- Factor by grouping. Group the first two terms and the last two terms. Pull a common factor out of each pair, then pull out the common binomial.\[ax^{2}+px+qx+c = (rx+s)(tx+u)\]
- Check by multiplying. Multiply your two binomials back out using FOIL (First, Outer, Inner, Last). You should get the original trinomial.\[(x+p)(x+q) = x^{2}+(p+q)x+pq\]
Worked example
Factor: 2x^2 + 7x + 3
Step 1: The three terms have no common factor, so we move on.
Step 2: Use the AC method. a x c = 2 x 3 = 6, and b = 7. We need two numbers that multiply to 6 and add to 7. Those are 1 and 6.
Step 3: Split the middle term: 2x^2 + 7x + 3 = 2x^2 + x + 6x + 3.
Step 4: Group and factor: (2x^2 + x) + (6x + 3) = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3).
Step 5: The factored form is (2x + 1)(x + 3).
Check: Use FOIL. (2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3. It matches.
Tips and common mistakes
- Write out all the factor pairs of c (or a x c) in a list before you guess. This way you do not miss one.
- If c is positive, the two numbers have the same sign. If c is negative, they have opposite signs.
- If c is positive and b is positive, both numbers are positive. If c is positive and b is negative, both are negative.
- Always pull out the common factor first. 6x^2 + 18x + 12 is much easier as 6(x^2 + 3x + 2).
- When a is not 1, the AC method (split and group) is more reliable than guessing.
- If no pair of whole numbers works, the trinomial cannot be factored. Do not make up factors.
- Always check with FOIL. The middle term is where sign mistakes hide.