A step-by-step order for factoring polynomials, with the special factoring formulas and an example for each method.
What does it mean to factor a polynomial?
Factoring a polynomial means rewriting it as a product of simpler polynomials. It’s multiplication run backwards: (x + 2)(x + 3) expands to x² + 5x + 6, so factoring x² + 5x + 6 gives (x + 2)(x + 3).
You need this skill for solving quadratic equations, simplifying rational expressions, and finding the zeros of a function. Learning how to factor polynomials comes down to two things: a short list of patterns and a sensible order for trying them.
How do you factor polynomials step by step?
Take out the greatest common factor, then count the terms and pick the method that fits that count. Repeat on each factor until nothing else breaks down.
- Factor out the GCF of all the terms, including any common power of x.
- Two terms: look for a difference of squares, or a sum or difference of cubes.
- Three terms: check for a perfect square trinomial, then use the ac method or trial and error.
- Four terms: try factoring by grouping.
- Look at each factor again to see if it factors further.
- Multiply your factors back out to check.
How do you factor out the greatest common factor?
Find the largest number and the highest power of each variable that divide every term, then write that outside a bracket. Divide each term by it to get what goes inside.
For 6x³ + 9x², the GCF of 6 and 9 is 3, and the lowest power of x is x². So the GCF is 3x². Dividing gives 2x and 3, and the result is 3x²(2x + 3).
Skipping this step is the most common reason a factored answer is marked incomplete. The guide on how to find the GCF covers the number part in more depth.
What are the special factoring formulas?
Four factoring formulas cover most special cases: difference of squares, the two perfect square trinomials, and the sum and difference of cubes.
How do you factor a trinomial like ax² + bx + c?
When a = 1, find two numbers that multiply to c and add to b. When a isn’t 1, use the ac method: find two numbers that multiply to a × c and add to b, split the middle term, and group.
For x² + 5x + 6, you need numbers that multiply to 6 and add to 5. That’s 2 and 3, so the answer is (x + 2)(x + 3).
For 6x² + 7x − 3, a × c = −18. The pair 9 and −2 multiplies to −18 and adds to 7. Rewrite the middle term: 6x² + 9x − 2x − 3. Group it as 3x(2x + 3) − 1(2x + 3), which becomes (3x − 1)(2x + 3). The article on how to factor trinomials has more practice with this method.
How do you factor a polynomial with four terms?
Split the four terms into two pairs, factor the GCF out of each pair, and if the brackets match, factor that bracket out.
Take x³ + 3x² − 4x − 12. The first pair gives x²(x + 3). For the second pair, take out −4 so the bracket matches: −4(x + 3). Now you have (x + 3)(x² − 4).
Don’t stop there. x² − 4 is a difference of squares, so the full answer is (x + 3)(x − 2)(x + 2). Grouping often leaves a factor that one of the special patterns can finish off.
What does “factor completely” mean?
It means no factor in your answer can be factored any further. You usually need two methods in a row to get there.
For 2x³ − 18x, the GCF is 2x, giving 2x(x² − 9). Then x² − 9 is a difference of squares, so the complete answer is 2x(x − 3)(x + 3). Stopping at 2x(x² − 9) leaves work undone.
Another example is 3x² + 12x + 12. Take out 3 to get 3(x² + 4x + 4), then spot the perfect square: 3(x + 2)². And x⁴ − 16 needs the difference of squares twice: (x² − 4)(x² + 4), then (x − 2)(x + 2)(x² + 4).
What are common mistakes when factoring polynomials?
The big three are forgetting the GCF, getting the signs wrong in a trinomial, and trying to factor a sum of squares as if it were a difference.
For signs, look at c first. If c is positive, both numbers share the sign of b. If c is negative, the numbers have opposite signs, and the larger one takes the sign of b. In x² − x − 12, c is negative and b is −1, so the pair is −4 and 3, giving (x − 4)(x + 3).
When you’re unsure, multiply your answer back out. (x − 4)(x + 3) expands to x² − x − 12, which matches. That check takes a few seconds and catches nearly every sign error.
How do you factor polynomials of degree 3 or higher?
Try the GCF and grouping first. If neither works, use the rational root theorem to find a root, divide it out with synthetic division, and factor the smaller polynomial that’s left.
The factoring calculator on this site follows the same order: GCF, special patterns, the ac method, grouping, and the rational root theorem for higher degree polynomials. It shows each step, so you can see which pattern it spotted and why.
How does factoring polynomials help you solve equations?
Once a polynomial equation is factored and set equal to zero, each factor gives you a solution. This works because a product can only be zero if one of its factors is zero.
Solve x² − x − 12 = 0. Factor the left side to (x − 4)(x + 3) = 0. Then either x − 4 = 0 or x + 3 = 0, so x = 4 or x = −3.
The same idea scales up. From x³ + 3x² − 4x − 12 = 0, the factored form (x + 3)(x − 2)(x + 2) = 0 gives three solutions: x = −3, 2, and −2. Make sure one side is zero before you start. Factoring x² − x = 12 as x(x − 1) = 12 doesn’t let you split it into separate equations.
Frequently asked questions
What is the first step in factoring polynomials?
Look for a greatest common factor and pull it out. Every other method gets easier once the GCF is gone, and leaving it in is the most common reason an answer is marked incomplete.
How do you know when a polynomial can’t be factored?
If there’s no GCF, it doesn’t match a special pattern, and no pair of integers works for the ac method, it’s prime over the integers. For a quadratic, a discriminant that isn’t a perfect square confirms it.
Can you factor a sum of squares like x² + 9?
Not over the real numbers. A sum of two squares has no real factors, so x² + 9 stays as it is. Only the difference, like x² − 9, factors.
What is the difference between factoring and expanding?
Expanding multiplies brackets out into a sum of terms. Factoring goes the other way and turns a sum back into a product. Each one checks the other.
Does the order of the factors matter?
No. (x + 3)(x − 2) and (x − 2)(x + 3) are the same, because multiplication is commutative.
Which factoring formulas should I memorize?
Difference of squares and the two perfect square trinomials come up most often. Sum and difference of cubes appear less, but they’re hard to work out on the spot, so they’re worth learning too.