Factoring Polynomials Calculator
Give a class the polynomial x⁴ − 16 and most of them will hand back (x² − 4)(x² + 4). It is not wrong. It is just not finished — and the mark scheme says factor completely. That one missing step, the second difference of squares hiding inside the first, costs more marks each year than any other mistake in the topic.
Factoring is not really a calculation. It is a decision: which method does this polynomial want? Once you know that, the algebra takes twenty seconds. The factoring polynomials calculator above answers that question out loud. It factors completely, and beside every step it names the rule it used, so you can see the reasoning rather than just the result.
The Order You Try Things In
This is the whole skill, and it is rarely taught as a list. Work down it. Stop at the first one that fits.
- Greatest common factor. Every single time, before anything else.
- Two terms? Difference of squares, or sum or difference of cubes.
- Three terms? Perfect square trinomial first, then the AC method.
- Four terms? Grouping.
- Degree three or more, nothing fitting? Hunt for a rational zero and divide it out.
- Still stuck? It may well be prime. That is an answer, not a failure.
Then — and this is the part almost everybody skips — go back and check whether any bracket you produced can itself be factored again. That is what caught the x⁴ − 16 class.
Greatest Common Factor: Take It Out First
6x² + 9x is 3x(2x + 3). Both terms share a 3 and both share an x, so both come out. Students who skip this step still get there eventually, but they do it by wrestling a harder trinomial than they needed to.
Watch the sign. When the leading coefficient is negative, pull the minus out with the GCF: −2x² − 4x − 2 becomes −2(x + 1)², which is far easier to read than fighting the negatives all the way down.
Difference of Squares, and the Cubes
Three patterns worth knowing cold, because they turn up constantly:
| a² − b² | (a − b)(a + b) |
| a³ − b³ | (a − b)(a² + ab + b²) |
| a³ + b³ | (a + b)(a² − ab + b²) |
There is no such thing as a difference-of-squares rule for a sum of squares. x² + 4 does not factor over the rationals, and writing (x + 2)(x + 2) for it is a mistake I have seen in every set of papers I have ever marked. Multiply it back: you get x² + 4x + 4, which is not what you started with.
Factoring Trinomials: a = 1 and Everything Else
When the leading coefficient is 1, it is a guessing game with a very small search space. x² + 5x + 6: find two numbers multiplying to 6 and adding to 5. Two and three. Done: (x + 2)(x + 3).
When it is not 1, use the AC method and stop guessing. For 6x² + 7x − 3, multiply a by c: 6 × (−3) = −18. Now find two numbers with product −18 and sum 7 — that is 9 and −2. Split the middle term with them and group:
6x² + 9x − 2x − 3
= 3x(2x + 3) − 1(2x + 3)
= (2x + 3)(3x − 1)
The AC method is slower to write out than guessing, and it never fails. On a timed paper that trade is worth taking.
If trinomials are the whole of what you are working on, they have a page to themselves — the factoring trinomials calculator shows the search for the number pair itself, every factor pair of a×c tested against b.
Factoring by Grouping
Four terms, no common factor across all of them. Split into two pairs and factor each pair on its own:
x³ + 2x² + 3x + 6
= x²(x + 2) + 3(x + 2)
= (x + 2)(x² + 3)
If the two brackets do not come out identical, try reordering the terms before you give up on the method. Grouping fails as often from a bad pairing as from being the wrong tool.
When a Polynomial Will Not Factor
x² − 2 is prime over the rationals. So is x² + x + 1. Being unable to factor them is not a gap in your technique — there is genuinely nothing there to find.
For a quadratic you can settle it in one line with the discriminant, b² − 4ac. Negative means no real zeros at all. Positive but not a perfect square means the zeros are irrational, so the quadratic is prime over the rationals even though it does cross the axis.
Higher degrees are harder to be sure about, and this is where most calculators quietly give up: they return the polynomial unchanged and let you assume they found nothing. This one distinguishes between could not find a factorisation and there is no factorisation, and it only says the second when it has been established.
How to Use This Factoring Polynomials Calculator
Type the polynomial and press =. Terms can be in any order, brackets are expanded first, and fractions and decimals are both fine. The key icon switches to your own keyboard.
The band gives the factored form and the zeros underneath — because nobody factors for its own sake. Below it, every step labelled with the method that produced it, ending with the factors multiplied back out and matched against what you typed. If that check ever failed, the calculator would refuse to show an answer at all.
What This One Does Differently
The two tools that dominate these search results will factor for free and then charge you for the steps — exactly backwards for anyone trying to learn the topic. Another allows one free calculation per visit.
Three things here are deliberate. Every step is named, so you learn the decision and not just the outcome. The answer is proved, multiplied back out in front of you. And the arithmetic is exact throughout — whole-number fractions, never decimals — so a repeated factor is never lost to a rounding error.
If you need the zeros rather than the brackets, the rational zeros theorem calculator lists the candidates worth testing, and the algebraic multiplicity calculator shows how each zero shapes the graph.
Worked Examples
| Polynomial | Factored completely | Method |
|---|---|---|
| x² + 5x + 6 | (x + 2)(x + 3) | trinomial, a = 1 |
| 6x² + 7x − 3 | (2x + 3)(3x − 1) | AC method |
| 6x² + 9x | 3x(2x + 3) | GCF only |
| x⁴ − 16 | (x − 2)(x + 2)(x² + 4) | difference of squares, twice |
| 8x³ − 27 | (2x − 3)(4x² + 6x + 9) | difference of cubes |
| x³ + 2x² + 3x + 6 | (x + 2)(x² + 3) | grouping |
| x³ − x² − x + 1 | (x − 1)²(x + 1) | grouping, repeated factor |
| x⁴ + 4 | (x² − 2x + 2)(x² + 2x + 2) | no rational zero, still splits |
| x² − 2 | prime over the rationals | discriminant not a square |
That eighth row is worth a second look. x⁴ + 4 has no rational zero at all, so every method on the list above fails on it — and yet it factors. Textbooks call it the Sophie Germain identity and most calculators miss it entirely.
Frequently Asked Questions
How do you factor a polynomial completely?
What does it mean to factor a polynomial completely?
How do you factor a trinomial with a leading coefficient?
What is factoring by grouping?
Can every polynomial be factored?
Is this factoring polynomials calculator free?
Factoring rewards pattern recognition more than it rewards algebra. Learn the order — common factor, then pattern, then grouping, then zeros — and check every bracket you produce before you call it finished. Use the calculator above to see which rule applies while you are still building that instinct, and you will stop needing it sooner than you think.