Factoring Generator
Factoring is multiplication run backward, and the Factoring Polynomials lesson covers where every pattern comes from. This generator 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 \(x^{2}+bx+c\), find the larger root | \(x^{2}+7x+12 \to q=4\) |
| Medium | GCF of coefficients, difference of squares, perfect square trinomials | \(x^{2}-16 \to p=4\) |
| Hard | The ac method, or a GCF plus a trinomial | \(3x^{2}+11x+6\) |
Easy factors a trinomial with leading coefficient \(1\) as \((x+p)(x+q)\) and asks for the larger of the two numbers — the exact “find two numbers that multiply to \(c\) and add to \(b\)” 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.
Using the generator
Pick a difficulty and a question count, and a fresh set appears instantly. Every answer here is a whole number.
- 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
| Pattern | Statement |
|---|---|
| GCF | \(ab+ac = a(b+c)\) |
| Difference of Squares | \(a^{2}-b^{2} = (a+b)(a-b)\) |
| Perfect Square Trinomial | \(a^{2}+2ab+b^{2} = (a+b)^{2}\) |
| Trinomial, leading coefficient 1 | \(x^{2}+(p+q)x+pq = (x+p)(x+q)\) |
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 \(p\) and \(q\) is positive and the other negative — check the product condition, not just the sum.
Forgetting the GCF only asks about the coefficients. \(20x^{3}\) and \(8x^{2}\) share an \(x^{2}\) too, but the question here is only about the numeric part, \(\gcd(20,8)=4\).
Mixing up which two numbers the ac method wants. They multiply to \(a \cdot c\), not just \(c\) — 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 Generator uses this exact skill to simplify fractions built from polynomials. For the full reasoning behind every pattern drilled here, see the Factoring Polynomials lesson.