Which method applies, and how to tell
Almost every factoring problem you will be set is one of five shapes. The hard part is not the working, it is recognising which shape you are looking at, so start by counting the terms.
| What you see | What it is | What to do |
|---|---|---|
| Anything at all | Common factor | Check this first, every time |
| Two terms, a minus, both squares | Difference of squares | a² − b² = (a − b)(a + b) |
| Two terms, both cubes | Sum or difference of cubes | a³ ± b³ = (a ± b)(a² ∓ ab + b²) |
| Three terms | Trinomial | The AC method |
| Four terms | Grouping | Split into pairs, factor each |
The first row is the one people skip and it is the one that costs marks. Pulling out a common factor makes everything after it smaller, and a problem that looks impossible often becomes a standard one the moment you do.
The AC method, which is the one worth learning properly
For a trinomial ax² + bx + c, multiply a × c, then find two numbers that multiply to that product and add to b. Take 2x² + 7x + 3:
- a × c = 2 × 3 = 6
- Two numbers that multiply to 6 and add to 7: 6 and 1
- Split the middle term with them: 2x² + 6x + x + 3
- Group the pairs: 2x(x + 3) + 1(x + 3)
- The bracket appears twice, so: (x + 3)(2x + 1)
That last step is the whole point of the method. When the same bracket comes out of both pairs, you are finished, and if it does not, you have either made an arithmetic slip or the polynomial does not factor.
Some things genuinely do not factor
This matters more than it sounds. x² + 1 does not factor. Neither does x² − x − 1, even though it looks like an ordinary trinomial. They are irreducible over the rationals, which means there is no way to write them as a product of polynomials with whole number coefficients.
Plenty of tools will hand you brackets anyway, full of decimals like (x − 1.618)(x + 0.618). Those are approximations to the roots, not a factorisation, and if you write them down as an answer it will be marked wrong. This page tells you when something will not factor, because that is the answer.
How the calculator actually does it
Underneath the named patterns sits one general tool: the rational root theorem. For a polynomial with whole number coefficients, any rational root written in lowest terms as p/q must have p dividing the constant term and q dividing the leading coefficient.
That turns an infinite search into a short list. For x³ − 6x² + 11x − 6, the constant is 6 and the leading coefficient is 1, so the only candidates are ±1, ±2, ±3 and ±6. Testing them finds 1, 2 and 3, and the polynomial is (x − 1)(x − 2)(x − 3).
Each root found peels off a factor by synthetic division, and the process repeats on what is left until nothing more divides. When the list runs out with nothing found, the polynomial is irreducible, and that conclusion is a proof rather than a guess.
Always multiply it back out
This page checks every answer by multiplying the factors back together and comparing to what you typed. It costs nothing and it is the reason you can trust the result.
Do the same by hand. Factoring goes wrong through sign errors far more often than through method errors, and expanding the brackets catches a sign error in about ten seconds. If it does not multiply back to what you started with, something is wrong, and you have found it before your teacher did.
Numbers factor too
Enter a whole number instead and you get its prime factorisation, which is the same idea applied to arithmetic: break it down until every piece is something that cannot be broken further. 360 becomes 2³ × 3² × 5.
Both kinds of factorisation are unique. For numbers that is the fundamental theorem of arithmetic, and it is the reason the answer above is the answer rather than an answer. Two people factoring the same thing correctly will always agree, which is not true of most things in mathematics.
Use it to check yourself, not to replace yourself
The steps above are named the way your teacher names them for a reason. An exam asks for the working, not the answer, so the useful way to use this page is to factor the thing yourself first and then see whether the method you picked was the one that applied.
If your answer and this one disagree, expand both. One of them will fail to multiply back, and finding out which is a better lesson than either answer on its own.
For the greatest common factor of several numbers the GCF calculator shows both standard methods, and once something is factored, solving for x is usually the next step, since a product equals zero exactly when one of its factors does.