Factoring Trinomials: Learn the AC Method Step by Step

Algebra Made Friendly

Factoring Trinomials: The AC Method and Beyond

A reliable step-by-step strategy for turning three-term polynomials into useful products.

Factoring is the reverse of multiplying. Instead of expanding a product such as (x + 2)(x + 3), you start with a trinomial such as x² + 5x + 6 and discover the factors that created it. The AC method gives you a dependable way to factor many trinomials, especially when the coefficient of x² is not 1.

What is a trinomial?

A trinomial is a polynomial with three terms. In this lesson, the common form is ax² + bx + c, where a, b, and c are numbers and a is not zero.

The big idea behind the AC method

The first and last coefficients are the key. Multiply the coefficient of x² by the constant:

A × C = a × c

Then find two integers whose product is a × c and whose sum is b. Those two numbers let you split the middle term and factor by grouping.

Memory tip: AC means multiply A and C, then look for numbers that add to B.

The four AC-method steps

1. Multiply: Calculate a × c.
2. Find: Find two integers with product a × c and sum b.
3. Split: Rewrite bx as the two terms suggested by those integers.
4. Group: Factor by grouping, then check by multiplying.

Worked example 1: 6x² + 11x + 3

Step 1 — Multiply A × C: 6 × 3 = 18.

Step 2 — Find the pair: 2 × 9 = 18 and 2 + 9 = 11.

Step 3 — Split the middle term:

6x² + 2x + 9x + 3

Step 4 — Group and factor:

2x(3x + 1) + 3(3x + 1) = (3x + 1)(2x + 3)

Answer: 6x² + 11x + 3 = (3x + 1)(2x + 3)

Worked example 2: 8x² − 2x − 3

Multiply: 8 × (−3) = −24.

Find: 4 × (−6) = −24 and 4 + (−6) = −2.

Split and group:

8x² − 6x + 4x − 3 = 2x(4x − 3) + 1(4x − 3)

Answer: 8x² − 2x − 3 = (2x + 1)(4x − 3)

When the leading coefficient is 1

For x² + bx + c, the AC product is simply c. Find two numbers that multiply to c and add to b:

x² + 5x + 6 = (x + 2)(x + 3)

Because 2 × 3 = 6 and 2 + 3 = 5, the factors are easy to identify. The AC method still works; this is just the quicker version.

A case that does not factor over integers

Consider 2x² + 3x + 7. The AC product is 2 × 7 = 14. Integer factor pairs of 14 are 1 and 14 or 2 and 7, and none add to 3. Therefore, this trinomial does not factor into integer binomials.

Do not force a factorization when no pair meets both conditions. Depending on the course, you may leave the expression as is or use another method, such as the quadratic formula.

Common mistakes to avoid

• Using a + c instead of a × c: The AC step is multiplication.
• Ignoring signs: A negative product requires one positive and one negative integer.
• Forgetting the greatest common factor: Always check for a GCF before or after factoring.
• Skipping the check: Multiply the final binomials to confirm the middle and last terms.
• Stopping after splitting: Splitting the middle term is only useful when you complete the grouping step.

Quick practice

Try each problem before opening the answers. On a small phone screen, swipe horizontally inside any equation box if needed.

#	Factor the trinomial	Hint
1	3x2 + 10x + 3	A × C = 9
2	5x2 − 13x − 6	A × C = −30
3	x2 − 9x + 20	Look for two numbers with product 20
4	4x2 + 4x + 1	It may be a perfect square

Table tip: on a small phone screen, swipe horizontally inside the table to see every column.

Answers

1. 3x² + 10x + 3 = (3x + 1)(x + 3)
2. 5x² − 13x − 6 = (5x + 2)(x − 3)
3. x² − 9x + 20 = (x − 4)(x − 5)
4. 4x² + 4x + 1 = (2x + 1)²

Final takeaway

Multiply A and C, find two numbers that add to B, split the middle term, and factor by grouping. Then multiply your answer back out. That final check turns the AC method from a memorized trick into a reliable algebra habit.