Factoring trinomials: printable worksheets

Factor ax² + bx + c completely: a = 1, the ac method, a greatest common factor first, and prime trinomials.

Sample worksheet

The first 6 problems of a real worksheet, made the way the builder makes it. Open it in the builder to change the number of problems, the difficulty and the layout, then print.

12 x^{2} + 56 x + 9

6 x^{2} - x - 2

5 x^{2} + 18 x + 16

20 x^{2} + 29 x + 6

9 x^{2} - 36 x + 11

15 x^{2} - 7 x - 4

Show answers

(2 x + 9) (6 x + 1)

(2 x + 1) (3 x - 2)

(x + 2) (5 x + 8)

(4 x + 1) (5 x + 6)

(3 x - 11) (3 x - 1)

(3 x + 1) (5 x - 4)

Factoring a trinomial means rewriting a quadratic expression as a product of two simpler pieces, and it is one of the skills that Algebra 1 students lean on for the rest of the course. It shows up again when solving quadratic equations, simplifying fractions that contain variables, and graphing parabolas. Because it is a pattern-finding skill, it improves quickly with plenty of varied practice and an answer key to confirm each result.

This worksheet produces trinomials that factor completely. You can ask for the simplest case first, where the leading coefficient is one and the student looks for two numbers that multiply to the constant and add to the middle coefficient. When you are ready, switch to expressions with a different leading coefficient, which call for the ac method of splitting the middle term and grouping. A hard setting mixes both kinds and adds a common factor that must be pulled out before anything else, which is a step students often forget. You can also include a share of prime trinomials, which cannot be factored at all, so that students learn to say so instead of forcing an answer.

Study tips

Remind students that every answer can be checked by multiplying the factors back out. If the product does not match the original expression, a sign is usually the culprit.

Check for a greatest common factor before trying any other method.
Pay close attention to the signs of the two numbers you choose.
Change the variable letter now and then so the pattern, not the letter, is what sticks.

Open the sample in the builder to adjust the settings and print a fresh sheet with its answer key.

What students practise

I can factor a trinomial completely.

Students factor quadratic trinomials completely: a greatest common factor first, then the pair method or the ac method, and recognize prime trinomials.

Key words

trinomial: a polynomial with three terms
greatest common factor: the largest factor shared by every term
prime: cannot be factored over the integers

Common mistakes

Skips the greatest common factor.
Chooses a pair with the right product but the wrong signs.

Standards

Common Core State Standards this worksheet practises.

A-SSE.2: Use the structure of an expression to rewrite it. (unverified)
A-SSE.3a: Factor a quadratic expression to reveal its zeros. (unverified)

Before this: Two-step equations (Grade 7)
Next: Derivative rules: power and chain (Calculus I)