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Polynomial Evaluate Calculator

Horner nested multiplication for a real polynomial at a real x. Same engine as Polynomial Roots. Not factoring. Not a second engine. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 21/21 tests · Production surface contract 1/1 · v1.2.0
Input interpretation
Enter values to calculate.
Result
—
Model
p(x) of a real polynomial of degree 1–8 at a real x by Horner nested multiplication, with an engine-linked figure of p and the evaluation point.
Scope
The Result Card includes an engine-linked figure of p (marker is published x, value). Same Horner nesting as REST/MCP. Curve samples are not a REST field.
Verification
Engine tested · 21/21 tests · Production surface contract 1/1 · Specification checked · v1.2.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
7 golden · 6 boundary · 8 property · Production surface contract 1/1 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.2.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (4 capabilities; 161 remain CURRENT) @ 2026-09-21T10:26:02.216Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Hornerp(x) = ((cₙ x + cₙ₋₁) x + … ) x + c₀
Constant firstc0, c1, …, cn
iSame engine as /polynomial-roots. n multiplies for degree n. Float64. Not symbolic factoring. Not polynomial expansion. Not the quadratic formula page.

How to use

1

Enter coefficients and x

Constant first. 1 + 2x + 3x² is 1, 2, 3. All-roots stays at /polynomial-roots.

2

Read p(x)

Horner uses n multiplies. The figure shows p and the evaluation point. Cubic closed form stays at /cubic-equation.

Example calculations

Common configurations with formula and result.

ϟ

Quadratic at 2

1 + 2x + 3x²

coeffs 1, 2, 3 · x=2
17
ϟ

Root of a cubic

(x−1)(x−2)(x−3) at x=2

coeffs −6, 11, −6, 1 · x=2
0

Polynomial Evaluate calculator specification

Version 1.2.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
p(x) = c0 + c1 x + … + cn x^n is evaluated by nested multiplication ((cn x + c_{n-1}) x + … ) x + c0. This page is method=evaluate on math.polynomial.roots. Real x. Degree 1–8. Not a second engine. Not CAS factoring. Not expansion.
What it calculates
p(x) of a real polynomial of degree 1–8 at a real x by Horner nested multiplication, with an engine-linked figure of p and the evaluation point.
Inputs
  • coeffs
  • x
Outputs
  • value
  • degree
  • solver
  • n_mul
Formula
p(x) = ((cₙ x + cₙ₋₁) x + … ) x + c₀
Assumptions
  • The Result Card includes an engine-linked figure of p (marker is published x, value). Same Horner nesting as REST/MCP. Curve samples are not a REST field.
  • Real coefficients
  • Real x
  • Leading coefficient nonzero
  • Degree 1–8
  • Float64
  • Not a CAS
Units
  • dimensionless
Boundary conditions
  • missing x → MISSING_REQUIRED_INPUT
  • leading coefficient 0 → INVALID_INPUT
  • degree 0 or one coefficient → INVALID_INPUT
  • degree > 8 → VALUE_OUT_OF_RANGE
  • unknown method (including factor, expand, cas) → INVALID_MODE
Example
coeffs=1,2,3 x=2 → 17
Validation cases

2 published on this page · 21/21 tests · Production surface contract 1/1 · View evidence

  • coeffs=1,2,3 x=2 → 17
  • method=factor coeffs=1,2,3 x=2 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.2.0

Other calculators in this family: Absolute Value Calculator, Absolute Value Inequality Calculator, Column Space Calculator, Condition Number Calculator, Cubic Equation Calculator, Least Squares Calculator, Left Nullspace Calculator, Linear Equation Calculator . Explore all Algebra & Equations.

→ Absolute Value Calculator Solve |ax + b| = rhs. Same engine as Linear Equation. Two solutions when rhs > 0; empty when rhs < 0. Not quadratic. Not a second engine. Not CAS. Runs locally. › → Absolute Value Inequality Calculator Solve |ax + b| ○ rhs for ○ in {<, ≤, >, ≥}. Same engine as Linear Equation. Inside the branches is an interval; outside is a union. Not quadratic. Not a second engine. Not CAS. Runs locally. › → Column Space Calculator Column space (image) of a small real square matrix from RREF pivot columns. Same engine as Linear System. Trivial image is {0}. Left-nullspace is a sibling. Runs locally. › → Condition Number Calculator κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ for a small real square matrix. Same engine as Linear System. Singular matrices return infinite. Not a 2-norm product. Runs locally. › → Cubic Equation Calculator Cardano / trigonometric closed form for a real cubic. Same engine as Polynomial Roots. Three real roots or one real plus a conjugate pair. Not Ferrari. Not a second cubic engine. Runs locally. › → Least Squares Calculator Linear least squares for a small real square system. Full rank uses QR; rank-deficient uses SVD. Same engine as Linear System. Not Statistics regression. Runs locally. › → Left Nullspace Calculator Left-nullspace (cokernel) of a small real square matrix as ker(Aᵀ). Same engine as Linear System. Trivial left kernel is {0}. Not Jordan form. Runs locally. › → Linear Equation Calculator Solve ax + b = rhs for x. Identity and inconsistent cases when a = 0. Inequality /linear-inequality. Absolute value /absolute-value. Not the quadratic formula. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second polynomial engine?

No. It is method=evaluate on math.polynomial.roots, the same engine as /calc/math/polynomial-roots. All-roots and Cardano cubic stay on that engine.

Do you factor or expand?

No. Symbolic CAS factoring and expansion are out of scope. This page only evaluates p(x) by Horner.

Why Horner instead of summing c_k x^k?

Nested multiplication uses n multiplies for degree n and is the numerically preferred evaluation of a monomial basis polynomial in Float64.