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Math & Statisticsalgebra

Polynomial Calculator

Evaluate a polynomial and its derivative at a point, with any integer roots found by search. Evaluation uses Horner, which needs far fewer multiplications and loses less precision than raising powers.

Also called: evaluate polynomial, polynomial roots.

For x cubed minus 6x squared plus 11x minus 6, enter 1, -6, 11, -6.

Value at x
0

The polynomial is 1x^3 - 6x^2 + 11x - 6. At x = 2 it evaluates to 0, and its derivative there is -1.

Derivative at x
-1
Degree
3
The polynomial
1x^3 - 6x^2 + 11x - 6
Integer roots found
1, 2, 3
Sum of the coefficients
0
How this is calculated
Method and background

How this is calculated

Horner rewrites the polynomial as nested multiplication, so a cubic needs three multiplications rather than six and accumulates less floating-point error. The derivative is formed by the power rule and evaluated the same way. Integer roots are found by testing divisors of the constant term, which is the rational root theorem in its most useful form, and it finds the roots that factor cleanly rather than all of them.

evaluated by Horner: repeatedly multiply by x and add the next coefficient
a_k
The coefficients, highest power first

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

a cubic with three integer roots

Coefficients, highest power first
1, -6, 11, -6
Evaluate at x
2

Value at x0

The polynomial factors as (x-1)(x-2)(x-3), so it vanishes at x = 2

Open this example

evaluated away from a root

Coefficients, highest power first
1, -6, 11, -6
Evaluate at x
0

Value at x-6

boundary: at zero the value is the constant term

Open this example

Method and limits

What it assumes

  • Coefficients given highest power first, which is the written order.

What it deliberately does not model

  • Only integer roots are searched for. Irrational and complex roots need a numerical method this does not run.
  • Degree is capped at what the coefficient list provides.

Formula version 1.0.0 · definition 1.0.0 · India · Report a problem with this calculator

Frequently asked questions

Why use Horner's method?
Fewer operations and better precision. Computing each power separately multiplies the rounding error along with the value, which shows up quickly on high degrees.