Advanced Math and Statistics

Polynomial Calculator

Enter polynomial coefficients to identify the degree, evaluate the polynomial at any x-value, calculate its derivative and indefinite integral, and estimate all real or complex roots.

Enter Polynomial Data

List coefficients from the highest power down to the constant term.

Polynomial coefficients

Example: 1, -6, 11, -6 represents x³ − 6x² + 11x − 6. Use commas or spaces between values.

Evaluate at x

The calculator uses Horner’s method to compute P(x) efficiently.

Your analysis will appear here

Enter coefficients and an x-value, then select “Analyze Polynomial.”

Polynomial

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Degree —

Leading coefficient —

P(x) —

Constant term —

Derivative

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Indefinite integral

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Estimated roots

What Is a Polynomial?

A polynomial is an algebraic expression made from variables, coefficients, and non-negative integer powers. A general polynomial in x can be written as P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ··· + a₁x + a₀.

Important polynomial parts

Coefficient: A number multiplying a variable term.
Degree: The highest exponent with a non-zero coefficient.
Leading coefficient: The coefficient of the highest-degree term.
Constant term: The term that does not contain x.
Root or zero: An x-value for which P(x) = 0.

How to Enter Coefficients

Enter coefficients in descending order of powers. Include a zero whenever a power is missing so the calculator can preserve the correct degree and term positions.

Input example

For 2x³ − 5x + 7, enter 2, 0, −5, 7. The zero represents the missing x² term.

Supported values

Positive and negative coefficients
Decimals and integers
Linear through higher-degree polynomials
Real and complex root approximations

How the Polynomial Calculator Works

Read and validate coefficients

The calculator separates your coefficient list, checks each value, and removes unnecessary leading zeros.

Construct the polynomial

Each coefficient is matched to the correct descending power of x to create a readable polynomial expression.

Evaluate P(x)

Horner’s method is used to evaluate the polynomial with fewer multiplications and good numerical efficiency.

Differentiate and integrate

The power rule creates the first derivative and indefinite integral, with the integration constant shown as C.

Estimate polynomial roots

Linear and quadratic roots are solved directly. Higher-degree roots are estimated numerically using an iterative complex-root method.

Polynomial Types by Degree

Degree	Name	General form	Maximum number of roots
0	Constant	a	Usually none
1	Linear	ax + b	1
2	Quadratic	ax² + bx + c	2
3	Cubic	ax³ + bx² + cx + d	3
4	Quartic	ax⁴ + bx³ + cx² + dx + e	4
n	Degree-n polynomial	aₙxⁿ + ··· + a₀	n, counting multiplicity

Frequently Asked Questions

Enter zero for the missing coefficient. For x³ − 4x + 1, use 1, 0, −4, 1 because the x² coefficient is zero.

Yes. When a polynomial has non-real roots, the results are displayed in a + bi form using numerical approximations.

Most higher-degree polynomials require numerical methods. Small rounding differences can appear because the roots are approximated iteratively.

P(x) is the value of the polynomial after the selected x-value is substituted into every occurrence of x.

Yes. The indefinite integral is displayed with + C because differentiating any constant produces zero.