Factoring Trinomials: Decomposition Method

Factor complex trinomials with leading coefficient. Math 10-C Alberta mathematics curriculum.

Lesson 4.5 of Factors and Roots in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• The Decomposition Method. Learn how to factor trinomials where the leading coefficient is greater than 1 using the six-step decomposition (split-the-middle) method, with a fully worked example.
• Working with Negative Terms. Apply decomposition to trinomials whose constant or middle term is negative, including how to handle a negative GCF when the binomials must match.
• Two Variables and GCF-First Problems. Extend decomposition to two-variable trinomials and learn why factoring out a GCF first can simplify the entire process.

Inside the lesson — a free preview

Why You Need a New Method

The Problem with a Leading Coefficient Greater than 1. In the previous lesson, you factored trinomials like x^2 + 5x + 4 by finding two numbers that multiply to 4 and add to 5.

That worked because the coefficient of x^2 was 1.

When the coefficient of x^2 is greater than 1 — for example, 6x^2 + 17x + 12 — you cannot simply find two numbers that multiply to 12 and add to 17.

Trying 12 and 5 gives (x + 12)(x + 5) = x^2 + 17x + 60, which is not 6x^2 + 17x + 12. You need a different approach.

The Decomposition Method — Six Steps

The Method: Decomposition (Splitting the Middle Term). For a trinomial ax^2 + bx + c:

Step 1: Compute the product a × c.

Step 2: Find two numbers that multiply to ac and add to b.

Step 3: Rewrite the middle term bx as the sum of two terms using those two numbers.

Step 4: Group the four terms into two pairs.

Step 5: Factor the GCF out of each pair. The binomial inside each set of brackets should be the same.

Step 6: Factor out the common binomial.

Key Tip. The binomials produced in Step 5 must match. If they don't, try splitting the middle term the other way — swap the order of your two numbers. If neither order works, double-check that your number pair is correct.

Worked example: Factor 6x^2 + 17x + 12

Example 1: Factor 6x^2 + 17x + 12. Step 1: Compute a × c.

a × c = 6 × 12 = 72

Step 2: Find two numbers that multiply to 72 and add to 17.

Factor pairs of 72: 1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12, 8 × 9.

8 + 9 = 17 — these are the numbers.

Step 3: Rewrite 17x as 8x + 9x:

6x^2 + 8x + 9x + 12

Step 4: Group into two pairs:

(6x^2 + 8x) + (9x + 12)

Step 5: Factor the GCF out of each group:

2x(3x + 4) + 3(3x + 4)

Both groups contain (3x + 4) — the binomials match.

Step 6: Factor out the common binomial (3x + 4):

Answer: (3x + 4)(2x + 3)

Verify: (3x + 4)(2x + 3) = 6x^2 + 9x + 8x + 12 = 6x^2 + 17x + 12 ✓

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