Two kinds of factoring, both shown with their working and checked by expansion
To factor is to write a thing as a product of smaller things, and the word names two skills that students constantly blur: breaking an integer into primes, and breaking a polynomial into factors. This page keeps them in separate tools and shows the working in both. Integers are trial-divided to their prime factorisation with multiplicities and a divisor count. Polynomials with integer coefficients are factored the disciplined way: common factor out first, then the rational root test and synthetic division for cubics, the ac method for quadratics, and in every case the claimed factorisation is verified by expansion in front of you, because a factorisation that does not expand back was never true.
Disclaimer: This calculator gives estimates for general information only. It is not financial, medical or other professional advice. Polynomial factoring here covers integer coefficients up to the stated degree; irrational and complex factorisations are reported through the roots where relevant.
What this calculator does
The integer tool answers the arithmetic question completely: prime factorisation, each prime’s multiplicity, whether the number is prime, and how many divisors it has (computed from the multiplicities, not by listing). The polynomial tool answers the algebra question with its method printed: greatest common factor first (the step whose omission makes everything downstream harder), then for quadratics the ac method (split the middle term using the pair that multiplies to a·c and adds to b), and for cubics the rational root theorem, which limits the possible rational roots to factors of the constant over factors of the leading coefficient, so the search is finite and every candidate is genuinely tested by synthetic division. A polynomial that will not factor over the rationals is told so plainly, with its real roots given instead when they exist.
How to use it
- Coefficients are integers by design. Decimal or fractional coefficients should be cleared first (multiply through by the common denominator), which is the same preparation your course teaches, and the GCF step often does part of it for you.
- Signs matter and are inputs, not formatting. Enter the constant’s sign as part of its value: 6x² + 11x + 3 and 6x² + 11x − 3 are different factorisations entirely.
- Read the steps as the method, not the decoration. Each printed step (GCF, the ac pair, each candidate root tested) is the line a marker looks for; copying the shape of the working is rather the point of a page like this.
- Zero leading coefficient changes the degree. If a is 0 the polynomial is not the degree you selected, and the tool says so instead of factoring a fiction.
The method
- Integers: trial division by increasing primes to the square root of what remains; exponents recorded as multiplicities; divisor count = the product of (exponent + 1) over all primes; a remainder of the number itself above 1 at the end is prime.
- Quadratics: extract GCF g. Seek integers p, q with p·q = a·c and p + q = b, split the middle term, and factor by grouping; equivalently factor directly as (mx + r)(nx + s) with m·n = a. Difference of squares is the b = 0 special case the grouping finds on its own.
- Cubics: extract GCF, list rational candidates ±(factors of d)/(factors of a), test by synthetic division, and on a hit divide down to a quadratic and factor that. Result verified by expanding the product of factors back to the entered coefficients.
- Verification: every polynomial result is expanded symbolically and compared coefficient by coefficient with the input before it is shown as a factorisation.
Worked examples
360. Trial division: 360 = 2³ × 3² × 5. Multiplicities 3, 2 and 1 give (3+1)(2+1)(1+1) = 24 divisors, and 360 is composite and not squarefree (squares divide it), all of which the integer tool prints rather than leaving the factorisation to speak alone.
6x² + 11x + 3. GCF is 1. a·c = 18 and the pair that adds to 11 is 2 and 9: split as 6x² + 2x + 9x + 3, group: 2x(3x + 1) + 3(3x + 1) = (2x + 3)(3x + 1). The expansion check multiplies back to the entered coefficients exactly.
2x³ + 3x² − 11x − 6. Rational candidates include ±1, ±2, ±3, ±6 and their halves; testing x = 2 by synthetic division brings a zero remainder, leaving 2x² + 7x + 3, which factors as (2x + 1)(x + 3). Full result: (x − 2)(2x + 1)(x + 3), expansion-verified.
Reading your results
For integers, read multiplicities as the number’s anatomy: 2³ × 3² × 5 says 360 is divisible by 8, by 9, by 5, and by every combination product, which is what the divisor count counts. For polynomials, read the factorisation as the roots in disguise: each linear factor (x − r) is the root x = r wearing work clothes, and a quadratic factor that refuses to split further over the rationals carries either irrational or complex roots, which the tool reports numerically so the information is not lost. A “does not factor over the integers” verdict is a result, not a failure: most polynomials do not, and knowing it in one testable step is the value.
Limitations
- Degree three is the ceiling. Quartics factor by Ferrari’s method or numerically and are outside this page; so are polynomials in two variables and factorisation over finite fields, each its own discipline.
- Integer coefficients only. Coefficients with roots or fractions in them need clearing or a computer algebra system; the page declines rather than rounding your input into a different polynomial.
- Very large integers slow down. Trial division is honest and fast to a trillion; beyond that, modern factoring (Pollard’s rho and friends) is a subject of its own, and the tool limits its range instead of pretending.
- Multiplicity in polynomials is shown by repetition. A squared factor appears as the factor twice in the expansion check; in examination style you would write it with an exponent, and the tool’s steps name it when it occurs.
Frequently asked questions
What is the difference between factors and factorisation?
A factor is any piece of the product (3 is a factor of 360; (x − 2) is a factor of the cubic above), while the factorisation is the complete product with nothing left over and nothing further reducible in the chosen number system: primes for integers, linear and irreducible quadratic factors for polynomials. The qualifier matters, since x² − 2 refuses to factor over the rationals yet splits over the reals as (x − √2)(x + √2), and a careful answer names its system, as this page does.
How do you factor a quadratic quickly?
Common factor first, then the ac pair: find two numbers multiplying to a·c and adding to b, split the middle term on them, and group. When a = 1 it collapses to the familiar hunt for a pair multiplying to c and adding to b. Speed comes from the order: students who skip the GCF hunt for pairs that the common factor was hiding, and students who test pairs against a·c rather than c alone miss every case where the leading coefficient is doing real work, as 6x² + 11x + 3 shows.
What is the rational root theorem?
The guarantee that any rational root of an integer polynomial, written in lowest terms as p/q, has p dividing the constant term and q dividing the leading coefficient. Its gift is finiteness: instead of guessing numbers forever, you list the candidates (for the cubic above, twelve of them) and test each by synthetic division or substitution. A root found this way hands you a linear factor and a polynomial one degree smaller, which is exactly the cubic tool’s procedure.
Why verify a factorisation by expanding it?
Because expansion is the definition running backwards: if the product of your factors multiplies out to the original polynomial, coefficient for coefficient, the factorisation is certainly right, and if it does not, no confidence in the method rescues it. The check costs seconds and catches the sign slips and dropped terms that are factoring’s natural predators, which is why the tool performs it on every result rather than trusting its own working.
Related calculators
Method: prime factorisation by trial division with multiplicities and divisor counts; polynomial factoring by GCF extraction, the ac method for quadratics, and the rational root test with synthetic division for cubics, every result verified by expansion. Last reviewed: 15 October 2026.