Factoring Polynomials Practice Worksheet

Practice factoring trinomials, special products, and polynomial expressions with step-by-step solutions.

Try each problem on your own first. Then click Show solution to check your work. These problems are designed to build confidence gradually.

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Level 1: Basic Trinomials

Problem 1
\[ x^2 + 7x + 12 \]
Show solution

Find two numbers that multiply to \(12\) and add to \(7\).

Those numbers are \(3\) and \(4\).

\[ x^2 + 7x + 12 = (x+3)(x+4) \]
Problem 2
\[ x^2 - 9x + 20 \]
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Find two numbers that multiply to \(20\) and add to \(-9\).

Those numbers are \(-5\) and \(-4\).

\[ x^2 - 9x + 20 = (x-5)(x-4) \]
Problem 3
\[ x^2 + 11x + 24 \]
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Find two numbers that multiply to \(24\) and add to \(11\).

Those numbers are \(3\) and \(8\).

\[ x^2 + 11x + 24 = (x+3)(x+8) \]
Problem 4
\[ x^2 - x - 30 \]
Show solution

Find two numbers that multiply to \(-30\) and add to \(-1\).

Those numbers are \(-6\) and \(5\).

\[ x^2 - x - 30 = (x-6)(x+5) \]

Level 2: Leading Coefficient Greater Than 1

Problem 5
\[ 2x^2 + 7x + 3 \]
Show solution

This factors into two binomials.

\[ 2x^2 + 7x + 3 = (2x+1)(x+3) \]

Check:

\[ (2x+1)(x+3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3 \]
Problem 6
\[ 3x^2 - 14x + 8 \]
Show solution

We want factors that produce \(3x^2\), \(8\), and a middle term of \(-14x\).

\[ 3x^2 - 14x + 8 = (3x-2)(x-4) \]

Check:

\[ (3x-2)(x-4) = 3x^2 - 12x - 2x + 8 = 3x^2 - 14x + 8 \]
Problem 7
\[ 4x^2 + 12x + 9 \]
Show solution

This is a perfect square trinomial.

\[ 4x^2 + 12x + 9 = (2x+3)^2 \]

Check:

\[ (2x+3)^2 = (2x+3)(2x+3) = 4x^2 + 12x + 9 \]
Problem 8
\[ 6x^2 + 11x + 3 \]
Show solution

We need binomials that multiply back to \(6x^2 + 11x + 3\).

\[ 6x^2 + 11x + 3 = (3x+1)(2x+3) \]

Check:

\[ (3x+1)(2x+3) = 6x^2 + 9x + 2x + 3 = 6x^2 + 11x + 3 \]

Level 3: Special Factoring Patterns

Problem 9
\[ x^2 - 49 \]
Show solution

This is a difference of squares.

\[ x^2 - 49 = x^2 - 7^2 \] \[ x^2 - 49 = (x-7)(x+7) \]
Problem 10
\[ 9x^2 - 25 \]
Show solution

This is also a difference of squares.

\[ 9x^2 - 25 = (3x)^2 - 5^2 \] \[ 9x^2 - 25 = (3x-5)(3x+5) \]
Problem 11
\[ x^2 + 10x + 25 \]
Show solution

This is a perfect square trinomial.

\[ x^2 + 10x + 25 = (x+5)^2 \]
Problem 12
\[ 16x^2 - 24x + 9 \]
Show solution

This is a perfect square trinomial.

\[ 16x^2 - 24x + 9 = (4x-3)^2 \]

Check:

\[ (4x-3)^2 = 16x^2 - 24x + 9 \]

Need more help with factoring?

Factoring becomes easier when students learn to recognize patterns. If you want a full explanation before practicing, visit my factoring polynomials guide.

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