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Factoring Higher-Degree Polynomials Practice Problems

Practice factoring higher-degree polynomials completely — the sum and difference of cubes patterns, a GCF pulled out first, and a quartic that's quadratic in form — with instant feedback across three difficulty levels.

  1. 1.

    Factoring starts by pulling out the GCF, leaving . Find .

  2. 2.

    Factoring starts by pulling out the GCF, leaving . Find .

  3. 3.

    Factoring starts by pulling out the GCF, leaving . Find .

  4. 4.

    Factoring starts by pulling out the GCF, leaving . Find .

  5. 5.

    Factoring starts by pulling out the GCF, leaving . Find .

  6. 6.

    Factoring starts by pulling out the GCF, leaving . Find .

  7. 7.

    Factoring starts by pulling out the GCF, leaving . Find .

  8. 8.

    Factoring starts by pulling out the GCF, leaving . Find .

  9. 9.

    Factoring starts by pulling out the GCF, leaving . Find .

  10. 10.

    Factoring starts by pulling out the GCF, leaving . Find .
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MPhil in Mathematics, Founder of Mathovia
Updated

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Factoring Higher-Degree Polynomials Practice Problems

Sum and difference of cubes, grouping, and quadratic-in-form substitution are all explained in the Factoring Higher-Degree Polynomials lesson. This practice set checks each one the same way: find one specific constant from the factored form.

How the difficulty levels work

DifficultyWhat it drillsExample
EasySum or difference of cubes, find ,
MediumThe same, with a GCF pulled out firstFind the GCF or
HardA quadratic-in-form quartic, or a bigger cubes problemFour linear roots: ,

Easy factors directly, asking for , the constant in the binomial factor.

Medium adds a GCF in front of the cube, alternating between asking for that GCF and asking for — two separate, equally necessary numbers.

Hard alternates between a quadratic-in-form quartic that factors into four linear roots (asking for the larger one) and a bigger sum-or-difference-of-cubes problem.

Worked examples

One example at each difficulty level, solved the way the practice set expects.

Easy. Factor .

, so use : , with .

Medium. Factor .

Pull out the GCF first: . Then the sum of cubes: . The GCF is and .

Hard. Factor completely.

Treat it as a quadratic in : . Each factor is a difference of squares: , so the roots are and .

Using the practice problems

Pick your difficulty level and question count, and a fresh set of problems loads right away. Every answer here is a whole number — enter it exactly as computed.

  • Check Answers compares every answer at once and flags the problems that missed
  • Show Answers shows every correct answer immediately, useful when marking a printed set
  • Print Worksheet formats a printer-friendly page, handy for classroom or take-home use

The rules these practice problems drill

RuleStatement
Difference of cubes
Sum of cubes
Quadratic in formSubstitute , factor as a quadratic, then substitute back

See the Factoring Higher-Degree Polynomials lesson for the full technique and worked examples combining several steps.

Common mistakes to watch for

Mixing up the signs in the cubes patterns. The trinomial factor’s middle sign is always the opposite of the binomial’s, and its last term is always positive.

Forgetting to substitute back after quadratic-in-form factoring. The substitution variable is only a bookkeeping tool — the final answer is written in terms of the original variable.

Skipping the GCF check between steps. A GCF can appear inside a factor that wasn’t obvious until after an earlier factoring step already ran.

Stopping after the first factoring step. A quartic factored into two quadratics may still have more factoring left — check every resulting factor again.

Where to go next

For the trinomial techniques this lesson builds on, see the Factoring Quadratics Practice Problems and the Factoring by Grouping Practice Problems. For the full technique behind every problem here, see the Factoring Higher-Degree Polynomials lesson.

Frequently Asked Questions

Why does this practice set ask for one number instead of the full factorization?+

A complete factorization like (x-2)(x²+2x+4) is an expression, which can't be automatically checked as a single number. Every problem is built backward from that exact factored form, so asking for one of its constants is a reliable check.

How do I remember the sign pattern in the sum and difference of cubes?+

The binomial factor keeps the same sign as the original expression, and the trinomial factor's middle sign is the opposite, with its last term always positive — see the Factoring Higher-Degree Polynomials lesson for the full pattern and why it works.

What does 'quadratic in form' mean in the Hard problems?+

A quartic whose only exponents are 4, 2, and 0 behaves exactly like a quadratic once x² is substituted with a single variable, and it factors into four linear roots — ±p and ±q — once that substituted quadratic is factored.

Why does Medium sometimes ask for the GCF instead of the cube's constant?+

Pulling the GCF out first doesn't change what the cube's constant b is, but it's a separate, equally important number to identify correctly before the cubes pattern can even be applied.

Do I need Factoring Quadratics before this practice set?+

Yes for the quartic problems — factoring a quadratic-in-form expression is just factoring an ordinary quadratic with a substituted variable, so that skill needs to be solid first.

Do I need to sign up to keep generating problems?+

No limit whatsoever — generate as many fresh sets as you want, free and account-free.

More practice problems

Factoring Quadratics Practice Problems Factoring by Grouping Practice Problems Polynomials Practice Problems