Factoring Polynomials Practice Problems

Factoring is multiplication run backward, and the Factoring Polynomials lesson covers where every pattern comes from. This practice set is where that understanding turns into speed: unlimited fresh problems, each checked by asking for the one specific number that only comes out right if the factoring was done correctly.

How the difficulty levels work

Difficulty	What it drills	Example
Easy	Factor , find the larger root
Medium	GCF of coefficients, difference of squares, perfect square trinomials
Hard	The ac method, or a GCF plus a trinomial

Easy factors a trinomial with leading coefficient as and asks for the larger of the two numbers — the exact “find two numbers that multiply to and add to ” skill the lesson introduces first.

Medium rotates between three recognition skills: finding the GCF of two coefficients, spotting a difference of squares, and spotting a perfect square trinomial.

Hard covers the two-coefficient case: the ac method (splitting the middle term when the leading coefficient isn’t 1), and factoring out a monomial GCF before a trinomial pattern still applies underneath.

Worked examples

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

Easy. Factor .

and , so . The larger constant is .

Medium. Factor and .

is a difference of squares: . In , both terms share : .

Hard. Factor .

, and . Split and group: . Multiply back out to check: .

Using the practice problems

Set a difficulty and a question count, then a brand-new set is generated on the spot. Every answer here is a whole number.

Check Answers grades every entry immediately, calling out the ones to fix
Show Answers reveals the full key at once, helpful for grading work done away from the screen
Print Worksheet builds a tidy printable page, good for classroom handouts or homework

The rules these practice problems drill

Pattern	Statement
GCF
Difference of Squares
Perfect Square Trinomial
Trinomial, leading coefficient 1

See the Factoring Polynomials lesson for the full pattern table, including the ac method for a leading coefficient other than 1.

Common mistakes to watch for

Getting the sign of one number right but not both. For a trinomial with a negative constant term, one of and is positive and the other negative — check the product condition, not just the sum.

Forgetting the GCF only asks about the coefficients. and share an too, but the question here is only about the numeric part, .

Mixing up which two numbers the ac method wants. They multiply to , not just — forgetting to include the leading coefficient in that product is the most common ac-method mistake.

Assuming every trinomial factors over the integers. Every problem here is deliberately built so it does — but not every trinomial you’ll see elsewhere will.

Where to go next

Once these feel automatic, the Rational Expression Practice Problems uses this exact skill to simplify fractions built from polynomials. For the full reasoning behind every pattern drilled here, see the Factoring Polynomials lesson.

Frequently Asked Questions

Why do I only have to find one number instead of writing out the full factored form?+

Typing a full factored expression like "(x+3)(x+4)" can't be automatically checked the way a number can. Finding the larger root, the GCF, or one of the ac-method's split numbers still requires doing the real factoring — it's just checked by asking for the one piece of it that has a single correct value.

What does p ≤ q mean in the Easy and Hard problems?+

The trinomial factors as (x+p)(x+q) for two numbers p and q, and since either order gives the same product, the problem always asks for whichever one is larger (labeled q) to remove the ambiguity.

How does the GCF question work?+

It asks only for the greatest common factor of the numeric coefficients, ignoring the shared variable part — the same as asking for the GCF of two plain numbers, which is exactly the first sub-skill of factoring out a monomial GCF.

What are the "two numbers" in an ac-method problem?+

For ax²+bx+c, they're the two numbers that multiply to a·c and add to b — the same pair used to split the middle term before factoring by grouping. See the Factoring Polynomials lesson's Worked Example B for the full method.

Do I need Polynomials before this practice set?+

Yes — factoring is multiplying in reverse, so being comfortable multiplying binomials (FOIL) first makes recognizing the reverse pattern much easier.

Is this free to use as much as I want?+

No cap exists. Every set is created instantly, free of charge, with no sign-up involved.

( )Factoring Polynomials Practice Problems