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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)

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

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

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  1. The greatest coefficient factor is 3.
  2. The lowest power of x is x^2.

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)

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  1. Factor out 4x^2.

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)

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  1. Group the first two and last two terms.

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

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  1. Factor x, then difference of squares.

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

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

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  1. The zeros are 0, 3, and -4.

Answer: 3

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