Quick start
- List the coefficients from the highest power down, separated by commas. For 2x³ − 3x² − 11x + 6, type 2, -3, -11, 6.
- Put 0 in for any missing power. x³ − 8 becomes 1, 0, 0, -8.
- Read the common factor, the roots found and the final factored form.
How the factoring works
First the tool divides out the greatest common factor of all the coefficients. Then it uses the rational root theorem: any fraction p/q that is a root must have p dividing the constant term and q dividing the leading coefficient. That gives a short list of candidates to test. Each one that makes the polynomial equal zero gives a factor, and the polynomial is divided down to a smaller one.
Example: 2x³ − 3x² − 11x + 6
The constant 6 has divisors ±1, ±2, ±3, ±6, and the leading 2 has divisors 1 and 2, so the candidates include ±1/2 and ±3/2 as well. Testing shows x = 3, x = −2 and x = 1/2 are roots. The result is (x − 3)(x + 2)(2x − 1).
When it cannot factor further
Some pieces have no rational roots. x² + 1 and x² − 2 cannot be split using whole numbers or fractions, so they are left as they are. For the roots of such a quadratic piece, use the Quadratic Equation Solver.
Common questions
Can I type the polynomial itself instead of coefficients?
This tool takes coefficients. To expand or tidy an expression written out in x, use the Simplify & Expand Calculator.
Why is a quadratic factor left over?
Its roots are irrational or complex, so it cannot be split with whole numbers. That is still the fully factored form over the rationals.
Does it handle a negative leading coefficient?
Yes. It can pull out −1 along with the common factor.