Factoring Trinomials Calculator
Factor any trinomial ax² + bx + c with step-by-step solutions using the AC method. Shows every step, identifies roots, and detects prime trinomials.
Example Trinomials
Enter Coefficients
Enter values for ax² + bx + c
Trinomial: x² + 5x + 6
Step-by-Step Solution
Factored Form
Roots (zeros):
x = -3, -2
AC Method Steps
- 1. Multiply a x c
- 2. Find two numbers that multiply to a×c and add to b
- 3. Rewrite bx as the sum of those two terms
- 4. Factor by grouping the 4-term polynomial
Common Trinomial Factoring Examples
| Trinomial | a × c | Split b as | Factored Form |
|---|---|---|---|
| x² + 5x + 6 | 6 | 2 + 3 | (x + 2)(x + 3) |
| x² - 5x + 6 | 6 | -2 + (-3) | (x - 2)(x - 3) |
| x² - x - 6 | -6 | 2 + (-3) | (x + 2)(x - 3) |
| 2x² + 7x + 3 | 6 | 1 + 6 | (2x + 1)(x + 3) |
| 3x² - 10x - 8 | -24 | 2 + (-12) | (3x + 2)(x - 4) |
| 6x² + 11x - 10 | -60 | -4 + 15 | (2x + 5)(3x - 2) |
| x² + 2x + 5 | 5 | No integer pair | Prime (not factorable) |
The AC Method for Factoring Trinomials
The AC method (also called the product-sum method or split-the-middle method) is the most systematic approach for factoring trinomials of the form ax² + bx + c, particularly when the leading coefficient a is not equal to 1. While simple trinomials like x² + 5x + 6 can often be factored by inspection or trial-and-error, the AC method provides a reliable algorithm that works for any factorable trinomial.
The method name comes from the first step: multiplying A (the leading coefficient) by C (the constant term). This product, often called the "key number," is the target for your factor search. You then find two integers whose product equals this key number and whose sum equals the middle coefficient B. Once found, these two integers replace the middle term, converting the trinomial into a 4-term polynomial that can be factored by grouping.
When a = 1: The Simple Case
When the leading coefficient equals 1, factoring simplifies significantly. For x² + bx + c, you need two numbers that multiply to c and add to b. For x² + 5x + 6, find two numbers multiplying to 6 and adding to 5 - that is 2 and 3. The factored form is (x + 2)(x + 3). This shortcut works because with a = 1, the AC method product a × c equals c, and the grouping steps simplify to just reading off the factor pairs.
Recognizing Special Forms
Some trinomials have special forms worth recognizing. A perfect square trinomial (a² + 2ab + b² = (a+b)²) factors into a repeated binomial. The difference of squares (a² - b²) isn't technically a trinomial but often appears when the middle coefficient is zero. Recognizing these patterns before applying the AC method can save time. For example, 4x² + 12x + 9 is a perfect square (2x + 3)², and x² - 9 factors immediately as (x + 3)(x - 3).
Prime Trinomials and the Discriminant
Not every trinomial can be factored over the integers. A prime trinomial is one where no integer factor pair satisfies the conditions. The quickest way to check before spending time searching for factors is to compute the discriminant: b² - 4ac. If the discriminant is a perfect square, the trinomial factors over integers. If it is zero, the trinomial is a perfect square. If it is negative, the trinomial has no real roots. If it is a positive non-perfect square, the trinomial has irrational real roots but no integer factoring.