How to Factor a Trinomial: Step by Step

What a trinomial is, and what factoring does to it

A trinomial is a polynomial with three terms — x² + 5x + 6 is the standard example. Learning how to factor a trinomial means learning to run multiplication backwards: instead of expanding (x + 2)(x + 3) into x² + 5x + 6, you start from the trinomial and recover the two binomials.

That matters for one specific reason. A product equals zero only when one of its factors equals zero. So the moment a quadratic is in factored form, its roots are readable straight off the page — no formula, no rearranging. Factored form is not a tidier way of writing the same thing; it is the form that answers the question.

Step 1: always take out the common factor first

Before anything else, check whether every term shares a factor. In 2x² + 10x + 12, each coefficient is even, so 2 comes out of the whole expression and leaves a simpler trinomial behind.

Skipping this step is the single most common way to get a wrong answer that still looks plausible. The greatest common factor never disappears — if you leave it inside, the numbers you are hunting for get larger and often stop being whole numbers at all.

Step 2: when the leading coefficient is 1

For x² + bx + c, you need two numbers that multiply to c and add to b. That is the whole method. For x² + 5x + 6 the pair is 2 and 3, because 2 × 3 = 6 and 2 + 3 = 5, so the factored form is (x + 2)(x + 3).

The signs follow a pattern worth memorising, because it narrows the search immediately:

So for x² − 5x + 6 the pair is −2 and −3, not 2 and 3. The constant is positive, which forces both numbers to share a sign, and the negative middle term makes that sign negative.

Step 3: when the leading coefficient is not 1

For 6x² + 7x + 2 there is no pair that simply multiplies to 2 and adds to 7. Here you use the AC method: multiply a by c, find a pair that multiplies to that product and adds to b, then split the middle term and factor in two groups.

The step people distrust is the third one, where a single 7x becomes 3x + 4x. It is allowed because 3x + 4x is 7x — nothing has changed, the expression has only been rewritten in a form that groups.

The shortcut: differences of squares and perfect squares

Two patterns are worth recognising on sight, because they skip the search entirely.

• A difference of squares has no middle term: x² − 9 factors as (x + 3)(x − 3). Anything of the form a² − b² factors as (a + b)(a − b).
• A perfect square trinomial has two identical factors: x² + 6x + 9 is (x + 3)². You can spot it when the first and last terms are perfect squares and the middle term is twice the product of their roots.

Note that a sum of squares, x² + 9, does not factor over the real numbers at all. That asymmetry catches people out constantly.

How to tell in advance whether a trinomial will factor

Not every trinomial factors into whole numbers, and no amount of searching will change that. The discriminant tells you before you start: compute b² − 4ac.

• If it is a perfect square, integer factors exist and the search will succeed.
• If it is positive but not a perfect square, the roots are irrational — use the quadratic formula instead.
• If it is negative, there are no real roots and the trinomial does not factor over the reals.

For x² + x + 1 the discriminant is 1 − 4, which is −3. That expression is irreducible, and knowing so in one line is faster than trying every pair of numbers.

Checking your answer, always

Expand the factors back. If multiplying them out does not reproduce exactly what you started with, something is wrong — usually a sign. This check takes ten seconds and catches essentially every mistake in the process.

The Factoring Calculator on this site runs that check internally before it shows an answer, and refuses to show a factorisation that does not expand back correctly.

Questions about how to factor a trinomial