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Polynomial Evaluator

Calculate P(x) from coefficients and a chosen x value

Polynomial coefficients
x =

Polynomial Evaluation Formula

P(x)=aₙxⁿ+aₙ₋₁xⁿ⁻¹+...+a₀

Polynomial evaluation means substituting a specific number for x and simplifying the expression until a single value remains. For coefficient input, each coefficient belongs to a descending power of x, from the highest degree down to the constant term. This calculator also uses Horner method, which rewrites the same calculation as repeated multiplication and addition, making long polynomials faster and easier to evaluate accurately.

Note: Coefficients are entered from highest degree to constant term.

Evaluation Methods

Polynomial evaluation finds the output value of a polynomial for a specific input. It can be done by direct substitution or by Horner method, which is faster and cleaner for long coefficient lists.

Substitution

Replace x by the chosen value and simplify.

Horner Method

Repeatedly multiply by x and add the next coefficient.

Function Value

The final result is written as P(x), such as P(2)=9.

Remainder Link

P(c) is also the remainder when dividing by x-c.

💡 Example: For P(x)=2x³-3x²+5 and x=2, P(2)=16-12+5=9.

Applications

Function ValuesRemainder TheoremGraphing

Frequently Asked Questions

What is a polynomial evaluator?
A polynomial evaluator calculates P(x) by substituting a chosen x value into a polynomial.
How do I enter the polynomial?
Enter coefficients from highest degree to constant term, separated by commas.
What is Horner method?
Horner method evaluates a polynomial efficiently by repeated multiplication and addition.
Can x be negative or decimal?
Yes. The calculator accepts positive, negative, and decimal x values.
Where is polynomial evaluation used?
It is used in function values, graphing, remainder theorem checks, interpolation, and numerical methods.

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