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Factoring Higher-Degree Polynomials

A free Algebra II lesson from the “Polynomial Factorization” unit, with a worked example and practice problems including step-by-step solutions.

Higher-degree polynomial factoring often starts with a greatest common factor. Grouping can factor four-term polynomials by pairing terms with a shared binomial.

What you'll learn

Why it matters: Polynomial models of growth, volume sums, and root-finding for engineering equations rely on factoring beyond degree two. Grouping, GCFs, and sum-or-difference-of-cubes patterns each handle a different shape — recognize the form first, then apply the right tool.

Worked example

Problem. Factor x^3 + 3x^2 + 2x + 6 by grouping.

  1. Group: (x^3 + 3x^2) + (2x + 6).
  2. Factor each group: x^2(x + 3) + 2(x + 3).
  3. Factor the shared binomial: (x + 3)(x^2 + 2).

Answer: (x + 3)(x^2 + 2)

Practice problems

1. Factor the GCF from 6x^3 + 9x^2. Enter the GCF.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Factor the GCF from 6x^3 + 9x^2. Enter the GCF.
  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 coefficient factor is 3.
  4. The lowest power of x is x^2.
  5. Check the result by substituting or estimating: the response should match 3x^2 and make sense in the original problem.

Answer: 3x^2

2. Factor 4x^3 - 8x^2.

Choices: 4x^2(x - 2) · 4x(x - 2) · 8x^2(x - 1) · x^2(4x + 8)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Factor 4x^3 - 8x^2.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Factor out 4x^2.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 4x^2(x - 2)

3. Factor x^3 + 5x^2 + 2x + 10 by grouping.

Choices: (x + 5)(x^2 + 2) · (x + 2)(x^2 + 5) · (x - 5)(x^2 - 2) · (x + 10)(x^2 + 1)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Factor x^3 + 5x^2 + 2x + 10 by grouping.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Group the first two and last two terms.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (x + 5)(x^2 + 2)

4. Factor x^3 - 4x.

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

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

Answer: x(x - 2)(x + 2)

5. For x(x - 3)(x + 4) = 0, enter the positive zero.

Show solution
  1. Challenge: First identify exactly what the question is asking: For x(x - 3)(x + 4) = 0, enter the positive zero.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The zeros are 0, 3, and -4.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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