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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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published methods
Assurance
Core
Declared partition coverage
PASS · 4/4 declared partitions (all-roots, cardano, evaluate, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
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 · Specification checked · v1.2.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.2.1 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.discrete-algebra-cvp
Verification revision
2026-09-22.discrete-algebra-cvp · 3/3 property · digest 6a2ac2cde621
Legacy regression
21/21 tests · Production surface contract 3/3
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
7/7 golden · 5/5 CVP boundary · 10/10 invalid · 3/3 property · 4/4 cross-interface · 1/1 cross-calculator · 3/3 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
7 legacy golden · 6 legacy boundary · legacy regression suite · 7/7 oracle-backed golden · 10/10 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.2.1 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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 1.2.1 · 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 3/3 · 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
1.2.1

Other calculators in this family: Absolute Value Calculator, Absolute Value Inequality Calculator, Cubic Equation Calculator, Linear Equation Calculator, Linear Inequality Calculator, Polynomial Roots Calculator, Quadratic Formula Calculator, Quadratic Inequality 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. › → 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. › → 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. › → Linear Inequality Calculator Solve ax + b ○ rhs for ○ in {<, ≤, >, ≥}. Same engine as Linear Equation. Negative a flips the inequality. Not quadratic. Not a second engine. Runs locally. › → Polynomial Roots Calculator Find all roots of a real polynomial of degree 1–8. Closed form for linear, quadratic, and cubic (Cardano); Durand–Kerner otherwise. Cubic discovery /cubic-equation. Horner evaluation /polynomial-evaluate. Not the quadratic formula page. Runs locally. › → Quadratic Formula Calculator Solve ax² + bx + c = 0 with the quadratic formula. Inequality discovery /quadratic-inequality. Free online tool — no sign-up. Clear formulas and worked examples. Runs locally in your browser. › → Quadratic Inequality Calculator Solve ax² + bx + c ○ rhs for ○ in {<, ≤, >, ≥}. Same engine as Quadratic. Sign of a is the outside of the roots. Not cubic. Not a second engine. Not CAS. Runs locally. ›

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.