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Math Practice Generator

Factoring by Grouping Generator: Free Practice with Four-Term Polynomials

Practice factoring by grouping — splitting a four-term polynomial into matching pairs — plus the ac method for a trinomial with a leading coefficient other than 1, with instant feedback across three difficulty levels.

  1. 1.

    Factoring \(x^{3} + 5x^{2} + 3x + 15\) by grouping gives \((x+p)(x^{2}+q)\). Find \(p\).

  2. 2.

    Factoring \(x^{3} - 2x^{2} + 7x - 14\) by grouping gives \((x+p)(x^{2}+q)\). Find \(p\).

  3. 3.

    Factoring \(x^{3} + 2x^{2} + 7x + 14\) by grouping gives \((x+p)(x^{2}+q)\). Find \(p\).

  4. 4.

    Factoring \(x^{3} + x^{2} + 8x + 8\) by grouping gives \((x+p)(x^{2}+q)\). Find \(p\).

  5. 5.

    Factoring \(x^{3} + 8x^{2} - 9x - 72\) by grouping gives \((x+p)(x^{2}+q)\). Find \(p\).

  6. 6.

    Factoring \(x^{3} - 7x^{2} - 6x + 42\) by grouping gives \((x+p)(x^{2}+q)\). Find \(p\).

  7. 7.

    Factoring \(x^{3} + 6x^{2} + 7x + 42\) by grouping gives \((x+p)(x^{2}+q)\). Find \(p\).

  8. 8.

    Factoring \(x^{3} + 4x^{2} - 9x - 36\) by grouping gives \((x+p)(x^{2}+q)\). Find \(p\).

  9. 9.

    Factoring \(x^{3} - 4x^{2} + 7x - 28\) by grouping gives \((x+p)(x^{2}+q)\). Find \(p\).

  10. 10.

    Factoring \(x^{3} - 8x^{2} + 4x - 32\) by grouping gives \((x+p)(x^{2}+q)\). Find \(p\).
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Factoring by Grouping Generator

The Factoring by Grouping lesson covers pairing terms and the ac method in full; this generator drills the result directly, asking for one specific constant from the factored form every problem is built backward from.

How the difficulty levels work

DifficultyWhat it drillsExample
EasyGroup a four-term polynomial, find \(p\)\(x^{3}+4x^{2}+2x+8=(x+4)(x^{2}+2)\), \(p=4\)
MediumThe same, with negative constants allowed\(p\) or \(q\) can be negative
HardFind \(q\) instead, or the ac method’s smaller split numberBigger numbers throughout

Easy groups a four-term polynomial built from positive constants, asking for \(p\) — the constant in the linear factor.

Medium allows either constant to be negative, which changes the sign pattern of the four-term polynomial and tests the grouping process under a less friendly setup.

Hard alternates between asking for \(q\) — the constant in the quadratic factor — with bigger numbers, and the ac method’s two split numbers, asking specifically for the smaller one.

Using the generator

Pick a difficulty and a question count, and a fresh set appears instantly. Every answer here is a whole number, positive or negative — enter it exactly as computed.

  • Check Answers scores the set and flags exactly which problems need another look
  • Show Answers reveals every solution, useful for reviewing a paper attempt
  • Print Worksheet outputs a clean page for offline practice or classroom handouts

The rules this generator drills

RuleStatement
Grouping\(ax+ay+bx+by = a(x+y)+b(x+y) = (x+y)(a+b)\)
The ac methodFind two numbers multiplying to \(ac\), adding to \(b\)

See the Factoring by Grouping lesson for the full pairing technique and the complete ac-method walkthrough.

Common mistakes to watch for

Giving up when the first pairing doesn’t match. Rearranging the four terms — usually swapping the middle two — is a normal part of the process.

Finding two numbers that add to b but forgetting they must also multiply to ac. Both conditions are required at the same time in the ac method.

Factoring out a GCF with the wrong sign from the second pair. If the second pair starts negative, factoring out a negative GCF is usually what makes the two binomials match.

Stopping after factoring each pair separately. The matching binomial still needs one more factoring step to reach the final answer.

Where to go next

Once grouping feels automatic, the Factoring Quadratics Generator covers this ac method alongside every other trinomial-factoring technique. For the full pairing process and every worked example, see the Factoring by Grouping lesson.

Frequently Asked Questions

Why does this generator ask for p or q instead of the full factored form?+

A fully grouped answer like (x+3)(x²+4) is an expression, which can't be automatically checked as a single number. Every problem here is built backward from that exact factored form, so asking for one of its two constants — p or q — is a completely reliable check.

How do I find p or q without factoring the whole thing?+

Group the four terms into two pairs, factor the GCF out of each pair, and the constant left over in the resulting binomials is p or q, depending on which one the problem asks for. See the Factoring by Grouping lesson for the full step-by-step process.

Why does the ac-method subtype ask for the smaller number, not the larger?+

The Factoring Polynomials generator already asks for the larger of the two ac-method numbers, so this generator asks for the smaller one instead — same technique, full coverage of both numbers across the two generators.

Do the numbers in the four-term polynomial ever come out negative?+

Yes, starting at Medium — p and q can each be positive or negative, which changes the sign pattern of the whole four-term polynomial and is worth practicing directly.

Do I need Greatest Common Factor before this generator?+

Yes — grouping factors the GCF out of each pair separately, so being comfortable with that step first makes grouping much faster.

Is there a limit on how many problems I can generate?+

No. Sets are generated on demand and are unlimited and free, with no account required.

More practice generators

Greatest Common Factor Generator: Free GCF Practice Factoring Quadratics Generator: Free Practice for Every Trinomial Type Factoring Higher-Degree Generator: Free Practice with Cubes and Quartics