Polynomial Factoring Calculator

Factoring a polynomial by hand means guessing candidate roots, checking each one with synthetic division, and repeating on whatever's left — easy to lose track of once a root repeats or the constant term has a dozen divisors. This calculator runs the whole search for you: it pulls out the content, finds every rational root with its multiplicity, and tells you plainly whether anything's left over that doesn't factor over the rationals.

How to use it

1. Enter the polynomial's coefficients, highest degree first — for x³ − 3x + 2, enter 1, 0, -3, 2 (the missing x² term is a zero, and it matters: don't skip it).
2. Read off the content, the factored form, the root/multiplicity table, and whether the polynomial factors completely.

Only whole-number coefficients are supported — decimals and fractions aren't, since the method (the rational root theorem) is defined on integer coefficients.

The method

1. Content: divide out the GCD of every coefficient, leaving a primitive polynomial (content 1).
2. Rational root theorem: for a primitive polynomial with constant term a₀ and leading coefficient aₙ, every rational root is p⁄q in lowest terms, where p divides a₀ and q divides aₙ. Every such candidate is tested with exact-integer synthetic division for (qx − p).
3. Whenever a root is confirmed, it's divided out repeatedly (to catch multiplicity), and the search restarts from the smaller polynomial that's left — new candidates included, since deflating can expose roots that weren't visible in the original coefficients' divisors.
4. Once no candidate root divides evenly, whatever remains is reported as-is: either a bare ±1 (the polynomial factored completely) or a higher-degree piece with no rational roots.

FAQ

What does "no rational roots" mean for the leftover piece?

It means the search found no more p⁄q candidates that divide the polynomial evenly — not a formal proof that the piece can't be factored at all. A quadratic like x² + 1 has no real roots, and something like x² − 2 has real roots that are irrational (±√2); both come back as an unfactored remainder here, since neither has a rational root.

Why does the factored form sometimes start with a plain number?

That's the content — the GCD pulled out of every coefficient before searching for roots. 2x³ − 4x² − 2x + 4 factors as 2(x - 1)(x - 2)(x + 1), not (2x - 2)(x - 2)(x + 1), since 2 divides every term up front.

How is multiplicity shown?

Each distinct rational root gets one row in the table with its multiplicity — a double root shows multiplicity 2, and its factor in the factored form carries a matching ^2.

Related tools

To divide a polynomial by a specific x − a and check the remainder directly, see the Synthetic Division Calculator. To multiply factors back out into standard form, see the Expand Polynomials Calculator. For a quadratic specifically, the Quadratic Formula Calculator also handles irrational and complex roots that this tool's rational-only search won't find.

Privacy

All calculations run entirely in your browser — the coefficients you enter are never sent to a server.