CMClearMathAcademy

Factoring Foundations

A free Algebra I lesson from the “Exponents and Polynomials” unit, with a worked example and practice problems including step-by-step solutions.

Factoring rewrites a polynomial as a product. Start by checking for a greatest common factor, then look for patterns such as difference of squares or trinomials.

What you'll learn

Why it matters: Factoring reveals hidden structure in area expressions, break-even equations, and formulas that need to be solved in pieces.

Worked example

Problem. Factor 6x + 18.

  1. Find the greatest common factor of 6x and 18.
  2. The GCF is 6.
  3. Factor 6 out of each term: 6(x + 3).

Answer: 6(x + 3)

Practice problems

1. Factor 4x + 12.

Choices: 4(x + 3) · 4x(12) · x(4 + 12) · 2(2x + 12)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Factor 4x + 12.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. The greatest common factor is 4.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 4(x + 3)

2. Factor 3x^2 + 6x.

Choices: 3x(x + 2) · 3(x^2 + 6x) · x(3x + 6x) · 6x(3x + 1)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Factor 3x^2 + 6x.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Both terms share 3x.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 3x(x + 2)

3. Which is a difference of squares?

Choices: x^2 - 25 · x^2 + 25 · x^2 - 5x · x^2 + 10x + 25

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which is a difference of squares?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. It is x^2 minus 5^2.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x^2 - 25

4. Factor x^2 - 16.

Choices: (x - 4)(x + 4) · (x - 8)(x + 2) · (x + 4)(x + 4) · (x - 4)(x - 4)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Factor x^2 - 16.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Difference of squares: x^2 - 4^2.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (x - 4)(x + 4)

5. Factor x^2 + 7x + 12.

Choices: (x + 3)(x + 4) · (x + 2)(x + 6) · (x - 3)(x - 4) · (x + 1)(x + 12)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Factor x^2 + 7x + 12.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 3 and 4 multiply to 12 and add to 7.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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

Practice this interactively with instant feedback and an AI tutor.

Practice Factoring Foundations Take the free placement check

More Algebra I lessons