Horner's Method Calculator

About this calculator

Horner’s method (also called Horner’s scheme or nested multiplication) evaluates a polynomial at a given x value far more efficiently than computing each term’s power directly. A polynomial like a₃x³ + a₂x² + a₁x + a₀ is rewritten in nested form as ((a₃x + a₂)x + a₁)x + a₀: starting from the highest-degree coefficient, each step multiplies the running result by x and adds the next coefficient down, working through the coefficients highest-degree first.

This nested form needs only n multiplications and n additions to evaluate a degree-n polynomial, versus the roughly n(n+1)/2 multiplications a naive term-by-term approach requires when each power of x is computed separately — making Horner’s method both faster and less prone to floating-point rounding error. It’s the standard technique behind synthetic division and root-finding algorithms in numerical computing, and it’s how calculators, compilers, and scientific software libraries evaluate polynomials internally. This calculator takes a list of coefficients (highest degree first) and an x value and evaluates the polynomial via Horner’s method.