To factor a trinomial of the form x² + bx + c, find two numbers that multiply to c and add to b, then write the answer as two binomials. For x² + 7x + 12, the pair is 3 and 4 (3 × 4 = 12 and 3 + 4 = 7), so x² + 7x + 12 = (x + 3)(x + 4). When the x² term has a coefficient other than 1, the AC method extends the same idea.
Factoring is FOIL in reverse. Multiplying (x + 3)(x + 4) gives x² + 4x + 3x + 12, and the middle terms 4x and 3x combine into 7x. Factoring asks which two numbers produced that combination.
Step 0: Factor out the GCF
Always check for a common factor first.
3x² − 12x − 36 = 3(x² − 4x − 12)
Then factor the trinomial inside: −6 × 2 = −12 and −6 + 2 = −4
= 3(x − 6)(x + 2)
If the leading coefficient is negative, factor out −1 too: −x² + x + 6 = −(x² − x − 6) = −(x − 3)(x + 2).
Factoring x² + bx + c
- List the factor pairs of c, including negative pairs.
- Find the pair whose sum is b.
- Write (x + p)(x + q).
x² − 2x − 15: factor pairs of −15 are (1, −15), (−1, 15), (3, −5), (−3, 5)
The pair adding to −2 is 3 and −5
x² − 2x − 15 = (x + 3)(x − 5)
Use the signs to narrow the search
| Sign of c | Sign of b | The two numbers are | Example |
|---|---|---|---|
| + | + | Both positive | x² + 7x + 12 = (x + 3)(x + 4) |
| + | − | Both negative | x² − 9x + 20 = (x − 4)(x − 5) |
| − | + | Opposite signs, larger one positive | x² + 4x − 21 = (x + 7)(x − 3) |
| − | − | Opposite signs, larger one negative | x² − 2x − 15 = (x + 3)(x − 5) |
The “diamond problem” taught in many middle schools is this same search drawn as a diagram, with the product on top and the sum on the bottom. The diamond problem solver finds the pair for any product and sum, and the factors calculator lists every factor pair of c.
Factoring ax² + bx + c: the AC method
When a ≠ 1, the product-sum search alone is not enough, because the leading coefficient changes the middle term. The AC method handles it systematically.
- Multiply a × c.
- Find two numbers that multiply to ac and add to b.
- Split the middle term into those two terms.
- Factor by grouping: factor the GCF out of each pair of terms, then factor out the common binomial.
6x² + 11x − 10: ac = 6 × (−10) = −60
Two numbers with product −60 and sum 11: 15 and −4
Split: 6x² + 15x − 4x − 10
Group: 3x(2x + 5) − 2(2x + 5)
Factor: (3x − 2)(2x + 5)
2x² − 5x − 12: ac = −24; numbers −8 and 3 (product −24, sum −5)
2x² − 8x + 3x − 12 = 2x(x − 4) + 3(x − 4) = (2x + 3)(x − 4)
The order in which you write the split terms does not matter; 6x² − 4x + 15x − 10 groups to the same answer.
Trial and error
With practice, many people skip the AC steps and test binomial pairs directly. For 2x² − 5x − 12, the first terms must be 2x and x, and the last terms multiply to −12. Try combinations until the outer and inner products add to −5x: (2x + 3)(x − 4) gives −8x + 3x = −5x. This is fast when a and c have few factors and slow when they have many, which is when the AC method pays off.
Special patterns
Recognizing these saves time.
Perfect square trinomials. When the first and last terms are perfect squares and the middle term is twice the product of their roots:
- x² + 10x + 25 = (x + 5)²
- 4x² − 12x + 9 = (2x − 3)², since 2 × 2x × 3 = 12x
Difference of squares. Not a trinomial, but closely related: x² − 49 = (x + 7)(x − 7). The middle term is “missing” because +7x and −7x cancel. The difference of two squares calculator covers this pattern.
Two variables. The same method works when the constant is a multiple of y²: x² + 5xy + 6y² = (x + 2y)(x + 3y).
Quadratic form. Higher powers can hide a trinomial: x⁴ − 5x² + 4 = (x² − 1)(x² − 4), and each factor is a difference of squares, giving (x − 1)(x + 1)(x − 2)(x + 2).
When a trinomial will not factor
Not every trinomial factors over the integers. Check the discriminant before you spend time searching:
| Discriminant | Factorable with integers? |
|---|---|
| A perfect square (0, 1, 4, 9, 16, …) | Yes |
| Positive but not a perfect square | No; roots are irrational |
| Negative | No; roots are complex |
For x² + 5x + 3, D = 25 − 12 = 13, so it is prime. For x² + 3x + 5, D = 9 − 20 = −11, also prime. To find the roots of a prime trinomial, use the quadratic formula; the quadratic formula calculator does it in one step, and how to solve quadratic equations compares the methods.
Always check with FOIL
Multiply your answer back out: First, Outer, Inner, Last.
(3x − 2)(2x + 5) = 6x² + 15x − 4x − 10 = 6x² + 11x − 10 ✓
If the middle term comes out with the wrong sign, swap the signs in your binomials. If it has the right size but the wrong placement, swap which number goes with which first term. The FOIL calculator expands any pair of binomials for you.
Common mistakes
- Forgetting the GCF, so the final answer is not completely factored.
- Matching the sum but not the product, or vice versa. Both conditions must hold.
- Sign errors with negative c. When c is negative, the two numbers must have opposite signs.
- Stopping early with quadratic-form trinomials whose factors factor again.
The GCF step itself is covered in how to find the GCF and LCM, and the GCF calculator finds the common factor of all three coefficients.
Frequently asked questions
How do you factor a trinomial when the leading coefficient is 1?
For x² + bx + c, find two numbers that multiply to c and add to b, then write (x + first)(x + second). For x² + 7x + 12, the numbers 3 and 4 multiply to 12 and add to 7, so it factors as (x + 3)(x + 4).
What is the AC method?
For ax² + bx + c with a ≠ 1, multiply a × c, find two numbers with that product and a sum of b, split the middle term into those two pieces, and factor by grouping. It turns a hard trinomial into two easy binomial steps.
How can I tell if a trinomial cannot be factored?
Calculate the discriminant b² − 4ac. If it is not a perfect square (0, 1, 4, 9, 16, …), the trinomial cannot be factored using integers and is called prime. x² + 5x + 3 has discriminant 13, so it is prime.
Should I always factor out the GCF first?
Yes. Pulling out the greatest common factor first makes the remaining numbers smaller and easier to work with, and a trinomial is not completely factored until the GCF is out. 3x² − 12x − 36 = 3(x² − 4x − 12) = 3(x − 6)(x + 2).
How do I check my factoring?
Multiply the factors back together with FOIL. If you get the original trinomial, the factoring is right. For (3x − 2)(2x + 5): 6x² + 15x − 4x − 10 = 6x² + 11x − 10.