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Algebra · Polynomials

Polynomial Evaluation, Horner & Division Calculator

Evaluate a polynomial with Horner’s method, divide it synthetically by x − c, and verify the remainder theorem in one connected calculation.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Real-coefficient polynomials through degree six and a linear divisor x − c

Horner form

P(x)=(((aₙx+aₙ₋₁)x+…)x+a₁)x+a₀

Synthetic division

P(x)=(x−c)Q(x)+R

Remainder theorem

R=P(c)P(c)=0 ⇔ x−c is a factor
Algebra · Polynomials

Evaluate and divide with Horner’s method

Inputs and displayed decimal results use at most three decimal places. Exact internal values are retained during calculation.

P(x) = (x−c)Q(x) + P(c)

Enter coefficients, an evaluation point, and the divisor parameter c.

Using the model

Symbols, assumptions and limitations

Interpretation notes

The calculator accepts real coefficients through degree six. Synthetic division is performed only for a linear divisor x − c. Horner evaluation and the remainder theorem use the same nested arithmetic.

Worked examples

See the method in practice

01

A known cubic root

For P(x)=x³−6x²+11x−6 and c=1, synthetic division gives quotient x²−5x+6 and remainder 0.

02

Nonzero remainder

If P(c) is not zero, c is not a root and the same value is the remainder after division by x−c.

Questions

Frequently asked

Why is Horner’s method useful?

It evaluates a degree-n polynomial using only n multiplications and n additions.

What does remainder zero mean?

It means x−c is a factor and c is a root of the polynomial.

Can I divide by a quadratic?

Not in this focused tool; synthetic division here uses a divisor of the form x−c.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax College Algebra — Dividing Polynomials

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology