Polynomial Factorization
Factor a polynomial into linear factors over the rationals using the rational root theorem.
About Polynomial Factorization
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
Worked example
With the default values below, Polynomial Factorization returns Factored Form of (x - 1)(x - 2)(x - 3). Change any input above and every figure here is recalculated in your browser.
Inputs used
- Coefficients (highest to lowest degree)
- 1,-6,11,-6
Results
- Factored Form
- (x - 1)(x - 2)(x - 3)
- Rational Roots
- 1, 2, 3
- Remaining Factor
Step by step
- 1Values used
- 2Rational Root Theorem
- 3Factored Form= (x - 1)(x - 2)(x - 3)
- 4Rational Roots= 1, 2, 3
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.