How do you factor a trinomial like x² + 5x + 6?

What Is a Trinomial?

A trinomial is a polynomial with exactly three terms. The kind you factor in algebra class is the quadratic ax² + bx + c: x² + 5x + 6 has a = 1, b = 5 and c = 6, while 2x² + 7x + 3 has a = 2. Higher powers like x³ + 2x² + 1 are also trinomials, but they follow different rules (difference of cubes, grouping), so this page focuses on the quadratic form you will meet on every test.

Factoring a trinomial means rewriting it as a product of two binomials that multiply back to the original: (x + 2)(x + 3) expands to x² + 5x + 6, so the factorization is reversible. Those two binomials are also the equation’s x-intercepts — the reason factoring is the fastest route to solving a quadratic. If you actually want the factors of a whole number like 36 rather than a polynomial, use our factor calculator instead.

How to Use This Trinomial Factor Calculator (2026)

1. Enter the three coefficients. Use a, b and c from ax² + bx + c — for x² + 5x + 6 type a = 1, b = 5, c = 6. Negatives and a zero b are allowed (x² − 9 → a = 1, b = 0, c = −9).
2. Click Calculate. You get the factored form in the banner, plus stats for the discriminant, both roots, any GCF pulled out, and whether the expression is factorable over the integers.
3. Follow the working. The steps show the GCF check, the a × c product, the pair that splits the middle term, the grouping, and an expansion check that multiplies back to your original expression.
4. Read the verdict. If no integer pair works, the calculator reports prime over the integers and still gives you the roots and discriminant so you know why.

How to Factor a Trinomial (ax² + bx + c)

Factoring a quadratic expression means rewriting it as a product of two binomials. Start by pulling out any greatest common factor from all three terms — 6x² + 9x becomes 3x(2x + 3) before anything else — then use the numbers you have left.

When a = 1 (x² + bx + c): find two numbers that multiply to c and add to b. For x² + 5x + 6, the pair 2 and 3 multiplies to 6 and adds to 5, so the answer is (x + 2)(x + 3).

When a ≠ 1 (the ac method): multiply a × c, find two numbers that multiply to that product and add to b, then split the middle term and factor by grouping. For 2x² + 7x + 3: a × c = 6, and 1 + 6 = 7, so rewrite and group:

Special Patterns: Difference of Squares and Perfect Square Trinomials

Two patterns are worth memorising — once you spot them, no pairing or grouping is needed:

• Difference of squares: a² − b² = (a + b)(a − b), so x² − 9 = (x + 3)(x − 3) and 4x² − 25 = (2x + 5)(2x − 5). It has only two terms — the “missing” middle term is the giveaway.
• Perfect square trinomial: a² ± 2ab + b² = (a ± b)², so x² + 6x + 9 = (x + 3)² and x² − 10x + 25 = (x − 5)². The two end terms are perfect squares, and the middle term is twice their product.
• Sum of cubes (a bonus pattern): a³ + b³ = (a + b)(a² − ab + b²) — x³ + 8 = (x + 2)(x² − 2x + 4). Sums of squares never factor over the reals.

When Terms Are Missing, Negative or Fractional

b = 0 (missing middle term): the expression is a difference or sum of squares: x² − 9 → a = 1, b = 0, c = −9 factors as (x + 3)(x − 3).

c = 0 (no constant): factor x straight out: x² + 5x = x(x + 5). The calculator treats the remainder as a linear binomial with the root at x = 0.

Negative c: the two binomial constants have opposite signs — one adds, one subtracts. x² − 5x − 6 pairs as −6 and +1, giving (x − 6)(x + 1).

Negative a: factor the minus sign out first: −x² + 4x + 21 = −(x² − 4x − 21) = −(x − 7)(x + 3), which the calculator reports with the negative GCF already pulled out.

If no integer pair exists, the trinomial is prime over the integers — it still has roots, but they are irrational or complex. To see those roots, graph the expression and read where it crosses the x-axis with our graphing calculator.

How to Factor on a Calculator (TI-84, Casio & Desmos)

For polynomials, a non-CAS graphing calculator such as the TI-84 factors by solving: use the quadratic formula x = (−b ± √(b² − 4ac)) / 2a to get the two roots, then write each root r as the factor (x − r). Alternatively graph y = ax² + bx + c and read the x-intercepts — intercepts of −2 and −3 mean the factors are (x + 2)(x + 3). CAS models (TI-89, TI-Nspire CAS, Casio ClassPad, HP Prime) add a true factor() command that returns the factored form directly. You can also plot it in our graphing calculator and read the same intercepts on any device — our fraction calculator then reduces any fractional root you find. For factoring whole numbers instead of polynomials, use the prime factorization calculator.

Why Factoring Trinomials Matters

• Solving equations: x² + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0, so the solutions are x = −2 and x = −3 in one line — no formula needed.
• Graphing parabolas: the binomial factors give the x-intercepts directly, and the axis of symmetry sits halfway between them.
• Optimisation problems: projectile height, revenue and area models are quadratics — factoring finds when they hit zero (ground level, break-even).
• Algebra fluency: every calculus and physics course assumes you can factor ax² + bx + c on sight; the working here shows the same steps your textbook expects.

FAQ

Sources: Common Core State Standards for Mathematics (HSA.SSE.B.3, HSA.REI.B.4), NGA Center and CCSSO; National Council of Teachers of Mathematics, Principles to Actions; D. M. Rosen, Elementary Number Theory. This tool is for educational purposes only.