Polynomial Factorization
Factor a polynomial into linear factors over the rationals using the rational root theorem.
Inputs
e.g. '1,-6,11,-6' for x³−6x²+11x−6
Factored Form
(x - 1)(x - 2)(x - 3)
Rational Roots
1, 2, 3
Step by step
Values used
Coefficients (highest to lowest degree) = 1,-6,11,-6
Rational Root Theorem
If p/q is a rational root, then p | constant term and q | leading coefficient
Factored Form
= (x - 1)(x - 2)(x - 3)
Rational Roots
= 1, 2, 3
How it works
This calculator factors polynomials over the rationals using the Rational Root Theorem. It finds all rational roots p/q where p divides the constant term and q divides the leading coefficient, then deflates the polynomial via synthetic division. Note: only rational roots are found. Polynomials with purely irrational or complex roots cannot be factored by this method.
Formula
Rational Root Theorem
If p/q is a rational root, then p | constant term and q | leading coefficient
- p
- Numerator of rational root
- q
- Denominator of rational root
Frequently Asked Questions
What if no factorization is found?
The polynomial may only have irrational or complex roots, which this method cannot find. Try the cubic or quartic solver for exact roots.
Does this handle non-integer coefficients?
The rational root theorem works best with integer coefficients. Non-integer coefficients may yield inaccurate results due to floating-point precision.