Polynomial Evaluation Calculator
Evaluate a polynomial at a given x using Horner's method — enter the coefficients from highest to lowest degree.
How it works
Horner: P(x) = (…((cₙ·x + cₙ₋₁)·x + cₙ₋₂)·x + …)·x + c₀
A polynomial is evaluated most efficiently with Horner's method, which rewrites the sum of powers as nested multiplications. Instead of computing each xⁿ separately, it folds one coefficient in at a time — for a degree-n polynomial that is n multiplications and n additions, far fewer operations than the naive approach and with less rounding error. Enter the coefficients from the highest power down to the constant term.
Worked example
To evaluate 2x³ − 3x² + 5 at x = 2, enter coefficients 2, −3, 0, 5. Horner's method runs: 2 → 2·2 − 3 = 1 → 1·2 + 0 = 2 → 2·2 + 5 = 9. So P(2) = 9, matching 2·8 − 3·4 + 0 + 5 = 16 − 12 + 5 = 9.
Sources: Horner's method — Wikipedia
Frequently asked questions
How do I enter the coefficients?
List them from the highest degree down to the constant, separated by commas, and include zeros for any missing powers. For 2x³ − 3x² + 5 (which has no x¹ term) you enter 2, −3, 0, 5 — the 0 holds the place of the missing x term.
Why is Horner's method better than plugging in powers?
Evaluating each power xⁿ separately and adding takes about twice as many multiplications and accumulates more floating-point error. Horner's method uses only n multiplications and n additions for degree n, so it is both faster and more numerically stable — the standard way computers evaluate polynomials.
Can I use negative or decimal coefficients and x values?
Yes. Coefficients and the evaluation point x can be any real numbers, including negatives and decimals. The result is exact up to normal floating-point precision.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.