How do you factor a trinomial?

For x^2 + bx + c, find two numbers that multiply to c and add to b. For x^2 - 5x + 6: -2 times -3 = 6 and -2 + -3 = -5, so it factors as (x - 2)(x - 3). Always verify by expanding.

What does factoring mean?

Factoring rewrites an expression as a product of simpler expressions. Factoring x^2 - 9 gives (x - 3)(x + 3). It is the reverse of expanding, and it reveals the roots: each factor set to zero gives a solution.

Factoring Calculator

Factor ax² + bx + c into binomials. Enter whole-number coefficients, the calculator extracts any GCF first, then finds the integer factor pair with the cross-check shown.

How to use this calculator

Factoring reverses multiplication: it rewrites ax² + bx + c as (px + q)(rx + s). Enter whole-number coefficients and press Calculate. If a common factor exists it is pulled out first.

How it works

The cross-check: for (px + q)(rx + s), pr must equal a, qs must equal c, and, the key test, ps + qr must equal b (outer + inner). The calculator tries factor pairs until this check passes, or reports that no integer factorization exists.

(px + q)(rx + s) = ax² + bx + c where ps + qr = b

Worked example

Example: x² + 5x + 6

Step 1, GCF: coefficients 1, 5, 6 share no common factor.

Step 2, Pairs: factors of 1: (1, 1); factors of 6: (2, 3).

Step 3, Cross-check: (x + 2)(x + 3): 1×3 + 2×1 = 5 ✓.

Step 4, Answer: (x + 2)(x + 3). Expand to verify.

Common mistakes

Skipping the GCF: 2x² + 4x + 2 factors as 2(x+1)(x+1), missing the 2 is the most common incomplete answer.
Checking only pr and qs: getting a and c right is not enough; the middle term test ps + qr = b is what selects the right pair.
Forcing a factorization: some quadratics (like x² + 1) do not factor over integers at all. “Unfactorable” is a valid, complete answer.

When to use it

Solving quadratics by factoring (set each binomial to zero), simplifying rational expressions, finding x-intercepts quickly, and anywhere a quadratic needs to be rewritten as a product. New to the method? Read how to factor trinomials.

For a harder case worked end to end, see factoring x⁴−5x²+4, a biquadratic factored by substitution.

Frequently asked questions

What does it mean to factor a trinomial?

To rewrite ax² + bx + c as a product of binomials (px + q)(rx + s), reversing the multiplication you learned with FOIL.

What if the trinomial does not factor?

Then no integer pair satisfies the cross-check. The calculator says so plainly and reports the discriminant, the quadratic formula still solves it.

Should I factor out the GCF first?

Always. It simplifies the remaining search and is part of a complete answer, the calculator does this step automatically.

How is this different from prime factorization?

Prime factorization breaks a number into primes; this breaks a polynomial into binomials. See the prime factorization calculator for numbers.