Factoring Trinomials — Methods, Examples

Factoring Trinomials — Methods, Examples

TL;DR
Factoring trinomials rewrites ax² + bx + c as a product of two binomials by finding two numbers whose product is ac and whose sum is b. This article covers the three standard methods with examples and common mistakes.

Last updated on June 1, 2026
8 min read

Two Numbers That Decide Everything

Every quadratic trinomial that factors over the integers does so because two specific integers exist — two numbers whose product equals ac and whose sum equals b. This article addresses the bookkeeping around the search.

What "Factoring a Trinomial" Means

A trinomial is a polynomial with three terms, typically referred to in the algebra context as a quadratic trinomial: ax² + bx + c where a, b, and c are constants and a ≠ 0.

To factor the trinomial is to write it as a product of two binomials:

ax² + bx + c = (px + q)(rx + s)
where pr = a, qs = c, and ps + qr = b. When such integers exist, the trinomial is factorable over the integers. When they do not, the trinomial is called prime and must be solved by the quadratic formula instead.

Quick facts.

Method 1 — Sum-Product (When the Leading Coefficient Is 1)

For trinomials x² + bx + c:

  1. Find two numbers whose product is c and whose sum is b.
  2. Use those numbers as the constants in the binomial factors.

Example. Factor x² + 7x + 12. Two numbers with product 12, sum 7: 3 and 4. Answer: (x + 3)(x + 4).

Method 2 — The AC Method (When the Leading Coefficient Is Not 1)

For trinomials ax² + bx + c with a ≠ 1:

  1. Compute the product ac.
  2. Find two numbers m and n with mn = ac and m + n = b.
  3. Rewrite the middle term: ax² + mx + nx + c.
  4. Factor by grouping.
  5. The two groups will share a common binomial factor — extract it.

Example. Factor 2x² + 7x + 3. ac = 6. Two numbers with product 6, sum 7: 1 and 6. Rewrite: 2x² + x + 6x + 3 = (2x + 1)(x + 3).

Method 3 — The Perfect-Square Shortcut

If a and c are perfect squares and b = ±2√(ac), the trinomial is a perfect square:

a² + 2ab + b² = (a + b)²,
a² - 2ab + b² = (a - b)².

Example. x² + 10x + 25: Factor: (x + 5)².

Three Worked Examples of Factoring Trinomials

Quick.

Factor x² - 5x + 6.
Final answer: (x - 2)(x - 3).

Standard.

Factor 6x² + 11x - 10.
Final answer: (3x - 2)(2x + 5).

Stretch.

Factor 4x² - 12x + 9.
Final answer: (2x - 3)².

Why Factoring Trinomials Matters — Beyond the Quadratic

Factoring is essential for solving quadratics by the zero-product property, finding zeros of polynomial functions, graphing parabolas, simplifying rational expressions, and calculus.

Common Errors in Factoring Trinomials

1. Applying sum-product when a ≠ 1.

2. Sign errors when both numbers are negative.

3. Forgetting to check by multiplication.

4. Calling a non-factorable trinomial "wrong" instead of "prime."

The Mathematicians Who Shaped Factoring

Conclusion

Factoring trinomials rewrites ax² + bx + c as a product of two binomials. Always check by multiplying the factors back and when the discriminant is not a perfect square, use the quadratic formula.