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. In Exponents and Polynomials, students need more than a memorized rule: they need to recognize the structure, select a method, carry out the algebra cleanly, and interpret the answer in a graph, table, equation, or real context. The expanded practice now mixes skill fluency, transfer questions, and cumulative review so the lesson builds durable Algebra I readiness.
What you'll learn
- Factor out the greatest common factor
- Recognize difference of squares
- Factor simple trinomials
Worked example
Problem. Factor 6x + 18.
- Find the greatest common factor of 6x and 18.
- The GCF is 6.
- Factor 6 out of each term: 6(x + 3).
- Connect the result back to Factoring Foundations so the method and meaning are both clear.
Answer: 6(x + 3)
Practice problems
1. Factor 4x + 12.
Choices: 4(x + 3) · 4x(12) · x(4 + 12) · 2(2x + 12)
Show solution
- Warm-up: First identify exactly what the question is asking: Factor 4x + 12.
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- The greatest common factor is 4.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Identify the Algebra I structure before choosing a calculation.
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
- Warm-up: First identify exactly what the question is asking: Factor 3x^2 + 6x.
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- Both terms share 3x.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Identify the Algebra I structure before choosing a calculation.
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
- Warm-up: First identify exactly what the question is asking: Which is a difference of squares?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- It is x^2 minus 5^2.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Identify the Algebra I structure before choosing a calculation.
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
- Core Practice: First identify exactly what the question is asking: Factor x^2 - 16.
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- Difference of squares: x^2 - 4^2.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Identify the Algebra I structure before choosing a calculation.
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
- Core Practice: First identify exactly what the question is asking: Factor x^2 + 7x + 12.
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- 3 and 4 multiply to 12 and add to 7.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Identify the Algebra I structure before choosing a calculation.
Answer: (x + 3)(x + 4)
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