Last updated: 2026-06-06

Evaluating a polynomial at a point

A polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ assigns a number to every real x. Computing that number from scratch — raising x to each power, multiplying by the corresponding coefficient, and summing — requires O(n²) multiplications for a degree-n polynomial. Horner's method reduces this to exactly n multiplications and n additions by rewriting the same polynomial as a nested product:

$P(x) = (\cdots((a_n \cdot x + a_{n-1}) \cdot x + a_{n-2}) \cdot x + \cdots + a_1) \cdot x + a_0$

This is the algorithm used internally by every modern computer algebra system and numerical library.

This calculator takes the polynomial's coefficients from highest to lowest degree (for example, 1, -3, 2, 5 for x³ − 3x² + 2x + 5) and a value of x, then returns the degree, the formatted polynomial, and P(x), with a live graph of the curve centered on the evaluation point.

Horner's Method Step by Step

The algorithm maintains a running accumulator, initialized to the leading coefficient aₙ. For each subsequent coefficient it performs one multiply-then-add:

$r \leftarrow r \cdot x + a_{n-k} \quad \text{for } k = 1, 2, \ldots, n$

After n steps the accumulator holds P(x). This is optimal — no algorithm for polynomial evaluation can use fewer than n multiplications.

Worked example: P(x) = x³ − 3x² + 2x + 5 at x = 2 (coefficients: 1, -3, 2, 5)

So P(2) = 5 — confirmed by direct calculation: 8 − 12 + 4 + 5 = 5.

Why Horner's Method Is Faster

Direct evaluation computes each power of x separately: x² = x·x, x³ = x²·x, and so on. For degree n this costs up to n(n+1)/2 multiplications before the final summation. Horner's scheme reuses every intermediate product, bringing the count down to exactly n multiplications and n additions. For n = 20, that is 210 multiplications versus 20 — more than a 10× reduction.

Horner's method is also more numerically stable than the naïve approach: fewer intermediate large values means less catastrophic cancellation when coefficients differ in magnitude.

Entering Coefficients

List the coefficients from the highest-degree term to the constant term, separated by commas. Include a 0 for every missing power — omitting zeros shifts all remaining coefficients to the wrong degree.

The degree is always the number of entries minus one. A single entry is a degree-0 (constant) polynomial.

Real-World Uses

Polynomial evaluation is ubiquitous in computing and engineering:

• Computer graphics. Bézier curves and splines are parameterized polynomials; rendering one pixel on a cubic Bézier requires evaluating a degree-3 polynomial hundreds of times per frame.
• Signal processing. FIR filters are convolutions evaluated at each sample via polynomial multiplication in the Z-domain.
• Root finding. Newton's method for P(x) = 0 evaluates both P(x₀) and P′(x₀) at each iteration — both obtainable from a single Horner pass.
• Hash functions and checksums. Cyclic-redundancy checks (CRCs) evaluate a binary polynomial modulo a generator polynomial, which is a Horner loop over GF(2).
• Control systems. Transfer functions are rational polynomials in s (Laplace) or z (Z-transform); stability analysis requires evaluating the characteristic polynomial on the imaginary axis.

Relationship to Synthetic Division

Horner's method is algebraically identical to synthetic division — the shorthand algorithm for dividing P(x) by a linear factor (x − c). The remainder of that division equals P(c), and the quotient polynomial Q(x) satisfies P(x) = (x − c)Q(x) + P(c). Running one Horner pass therefore simultaneously produces P(c) (the remainder) and the coefficients of Q(x). Evaluating Q(c) gives P′(c), which is the basis for Newton's root-finding iteration.

Going Further

To compute P′(x) alongside P(x), use the Polynomial Derivative Calculator. To integrate the polynomial, use the Definite Integral of Polynomial Calculator. For solving P(x) = 0, use the Quadratic Equation Solver (degree 2) or Cubic Equation Solver (degree 3).