Factor Polynomial Calculator
This factor polynomial calculator splits a polynomial into factors: 2x³ + 2x² − 12x becomes 2x(x + 3)(x − 2) with the roots 0, −3 and 2. All steps and a check are shown.
Source: Wikipedia – Faktorisierung (Polynome). Updated: .
How it is calculated
Factoring means writing a polynomial as a product of simpler factors. The product 2x(x + 3)(x − 2) multiplies out to 2x³ + 2x² − 12x. From the factored form you can read off the roots directly, because a product is 0 when one factor is 0.
How the calculator proceeds
First it takes out the greatest common factor of the coefficients and the lowest powers of the variable. Then it looks for rational roots p/q: by the rational root theorem p divides the constant term and q divides the leading coefficient. Every root found gives a linear factor (q·x − p) by which the polynomial is divided, until no root is left. In the example, x² + x − 6 = (x + 3)(x − 2) remains besides the factor 2x.
Limits
The calculator factors over the rational numbers. A remainder such as x² + 1 has no real root and stays as it is. From degree 4 on, a remainder without a rational root can still split into quadratic factors, for instance x⁴ + 4 = (x² + 2x + 2)(x² − 2x + 2); it does not detect that.
- Exactly one variable is allowed; exponents from 0 to 20.
- Special products such as x² − 9 or x² + 6x + 9 are covered through the roots.
- The check multiplies the factors out again and compares with your input.
Frequently asked questions
How do you factor a polynomial?
Take out common factors, then look for roots and split off the linear factor (x − x₀) for each root x₀. x² − 5x + 6 has the roots 2 and 3, so (x − 2)(x − 3).
How do I find rational roots?
Try divisors of the constant term divided by divisors of the leading coefficient. For 2x² − 3x + 1 the roots are 1 and 1/2, giving (x − 1)(2x − 1).
What does a remainder that is not factored further mean?
It has no rational root. For x² + 1 there is no real root at all, so the factorization stays (x² + 1).
Sources and legal basis
- Wikipedia – Faktorisierung (Polynome)
- Wikipedia – Satz über rationale Nullstellen (rational root theorem)
- Wikipedia – Factorization of polynomials
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