How to Factor Trinomials? (+FREE Worksheet!)

Factoring trinomials is the reverse of multiplying binomials — your goal is to break a three-term polynomial back into two binomial factors. It is one of the most tested skills in Algebra 1 and is essential for solving quadratic equations. This guide covers both the simple case (leading \(\color{blue}{\text{ coefficient } = 1}\)) and the case when the leading coefficient is greater than 1, with four worked examples, two video lessons, and practice problems.

What Is Factoring a Trinomial?

A trinomial is a polynomial with exactly three terms, typically written as \(\color{blue}{\text{ ax }^{2} + \text{ bx } + c}\). Factoring it means writing it as a product of two binomials: \(\color{blue}{(\text{ px } + q)(\text{ rx } + s)}\). You can always check your answer by multiplying the factors back out using FOIL.

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Factoring When the Leading Coefficient Is 1

The “Find Two Numbers” Method

For trinomials of the form \(\color{blue}{x^{2} + \text{ bx } + c}\), find two integers that:

• Multiply to give the constant term \(\color{blue}{c}\), and
• Add to give the coefficient \(\color{blue}{b}\).

Then write the factored form as \(\color{blue}{(x + p)(x + q)}\) where p and q are those two numbers.

• \(\color{blue}{x^{2} + 7x + 12}\): need two numbers that multiply to 12 and add to 7 → 3 and 4.
  Factored: \(\color{blue}{(x + 3)(x + 4)}\)
• \(\color{blue}{x^{2} – 5x + 6}\): multiply to 6, add \(\color{blue}{\text{ to } -5}\) → −2 \(\color{blue}{\text{ and } -3}\).
  Factored: \(\color{blue}{(x – 2)(x – 3)}\)
• \(\color{blue}{x^{2} + 2x – 15}\): multiply \(\color{blue}{\text{ to } -15}\), add to 2 → 5 \(\color{blue}{\text{ and } -3}\).
  Factored: \(\color{blue}{(x + 5)(x – 3)}\)

Factoring When the Leading Coefficient Is Greater Than 1

The AC Method (Factoring by Grouping)

For \(\color{blue}{\text{ ax }^{2} + \text{ bx } + c}\) with \(\color{blue}{a \ne 1}\):

1. Multiply \(\color{blue}{a \times c}\) (the ac product).
2. Find two numbers that multiply to \(\color{blue}{\text{ ac }}\) and add to \(\color{blue}{b}\).
3. Rewrite the middle term using those two numbers.
4. Factor by grouping in pairs.

Example: Factor \(\color{blue}{2x^{2} + 7x + 3}\).

• \(\color{blue}{\text{ ac } = 2 \times 3 = 6}\). Find two numbers that multiply to 6 and add to 7: 6 and 1.
• Rewrite: \(\color{blue}{2x^{2} + 6x + x + 3}\)
• Group: \(\color{blue}{2x(x + 3) + 1(x + 3)}\)
• Factor: \(\color{blue}{(2x + 1)(x + 3)}\)

Step-by-Step Summary

1. Check if the leading coefficient is 1 or greater than 1.
2. If \(\color{blue}{a = 1}\): find two numbers with \(\color{blue}{\text{ product } = c}\) and \(\color{blue}{\text{ sum } = b}\); write as \(\color{blue}{(x + p)(x + q)}\).
3. If a > 1: compute ac, find two numbers with \(\color{blue}{\text{ product } = \text{ ac }}\) and \(\color{blue}{\text{ sum } = b}\), split the middle term, then factor by grouping.
4. Check your answer by FOILing the factors.

Watch: Factoring Trinomials — Concept Video

The Organic Chemistry Tutor gives a clear introduction to factoring both simple and leading-coefficient trinomials:

Factoring Trinomials — Worked Examples

Example 1: Factor \(\color{blue}{x^{2} + 7x + 12}\).

Find two numbers with product 12 and sum 7: 3 and 4.
Answer: \(\color{blue}{(x + 3)(x + 4)}\)
Check: \(\color{blue}{(x + 3)(x + 4) = x^{2} + 4x + 3x + 12 = x^{2} + 7x + 12}\) ✓

Example 2: Factor \(\color{blue}{x^{2} – 5x + 6}\).

Find two numbers with product 6 and \(\color{blue}{\text{ sum } -5}\): −2 \(\color{blue}{\text{ and } -3}\).
Answer: \(\color{blue}{(x – 2)(x – 3)}\)

Example 3: Factor \(\color{blue}{x^{2} + 2x – 15}\).

Product = −15, \(\color{blue}{\text{ sum } = 2}\): 5 \(\color{blue}{\text{ and } -3}\).
Answer: \(\color{blue}{(x + 5)(x – 3)}\)

Example 4: Factor \(\color{blue}{2x^{2} + 7x + 3}\).

\(\color{blue}{\text{ ac } = 6}\). Numbers with product 6 and sum 7: 6 and 1.
Rewrite: \(\color{blue}{2x^{2} + 6x + x + 3}\) → \(\color{blue}{2x(x + 3) + 1(x + 3)}\)
Answer: \(\color{blue}{(2x + 1)(x + 3)}\)

More Practice: Factoring Video

This Organic Chemistry Tutor video covers the full range of factoring techniques in detail:

Exercises for Factoring Trinomials

Factor each trinomial completely.

1. \(\color{blue}{x^{2} + 5x + 4}\)
2. \(\color{blue}{x^{2} – 8x + 15}\)
3. \(\color{blue}{x^{2} + x – 12}\)
4. \(\color{blue}{x^{2} – 9x + 20}\)
5. \(\color{blue}{2x^{2} + 5x + 2}\)
6. \(\color{blue}{x^{2} – x – 6}\)

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Answers

1. \(\color{blue}{(x + 1)(x + 4)}\)
2. \(\color{blue}{(x – 3)(x – 5)}\)
3. \(\color{blue}{(x + 4)(x – 3)}\)
4. \(\color{blue}{(x – 4)(x – 5)}\)
5. \(\color{blue}{(2x + 1)(x + 2)}\)
6. \(\color{blue}{(x – 3)(x + 2)}\)

Free Factoring Trinomials Worksheet

Ready to practice on your own? Download our free Factoring Trinomials worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Factoring Trinomials before a quiz or test.

Download Factoring Trinomials x2 bx c Worksheet

Frequently Asked Questions

How do you know which two numbers to choose?

List the factor pairs of the constant term (for \(\color{blue}{a=1}\)) or of ac (for a>1), then find the pair whose sum equals the coefficient b. Keeping signs in mind is essential — the product must be positive or negative to match the constant term.

What if the trinomial cannot be factored?

If no integer pair satisfies both conditions, the trinomial is prime over the integers. You can confirm this by computing the discriminant \(\color{blue}{b^{2} – 4\text{ ac }}\): if it is not a perfect square, the trinomial does not factor neatly over the integers.

Should you always check your factoring answer?

Yes. Multiply your binomials back using FOIL to verify you get the original trinomial. This takes only seconds and catches any sign errors.

Related Topics

• Multiplying Binomials
• Factoring Polynomials
• Perfect Square Trinomial
• Simplifying Polynomials
• Adding and Subtracting Polynomials