Factoring by Grouping Practice Problems

Pairing terms and the ac method both get a full treatment in the Factoring by Grouping lesson. Here, the check is direct: find one specific constant from the factored form each problem was built backward from.

How the difficulty levels work

Difficulty	What it drills	Example
Easy	Group a four-term polynomial, find	,
Medium	The same, with negative constants allowed	or can be negative
Hard	Find instead, or the ac method’s smaller split number	Bigger numbers throughout

Easy groups a four-term polynomial built from positive constants, asking for — 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 — the constant in the quadratic factor — with bigger numbers, and the ac method’s two split numbers, asking specifically for the smaller one.

Worked examples

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

Easy. Factor .

Group in pairs: . The shared binomial comes out: , so .

Medium. Factor .

, so . When the third term is positive, the second group factors out a positive number.

Hard. Factor by splitting the middle term.

, and . Write , then group: . The smaller split number is .

Using the practice problems

Choose how hard the questions should be and how many you want; the set is ready instantly. Every answer here is a whole number, positive or negative — enter it exactly as computed.

Check Answers runs through the whole set and marks which answers still need work
Show Answers displays every solution instantly, ideal for reviewing paper-based practice
Print Worksheet creates a print-friendly version, handy for handouts or offline work

The rules these practice problems drill

Rule	Statement
Grouping
The ac method	Find two numbers multiplying to , adding to

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 Practice Problems 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 practice set 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 practice set already asks for the larger of the two ac-method numbers, so this practice set asks for the smaller one instead — same technique, full coverage of both numbers across the two sets.

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 practice set?+

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

How many problem sets can I make in total?+

Not at all. You can generate unlimited sets for free, and no account is ever needed.

a·bFactoring by Grouping Practice Problems