What formats can I type in?

You can enter expressions like 3x^4 - 2x^2 + 5x - 7. Use ^ for exponents, and the variable x. Coefficients may be integers or decimals.

Does it factor every polynomial?

The tool performs exact rational factorization (via the Rational Root Theorem) and completes remaining quadratic factors. Fully general symbolic factorization is not guaranteed.

How are derivatives and integrals computed?

They follow power rules: (ax^n)' = a·n·x^(n-1) and ∫ax^n dx = a·x^(n+1)/(n+1) + C (n ≠ -1).

How are numeric plots drawn?

The function is sampled uniformly across the selected window and linearly connected. Axes and grid help interpret intercepts and turning points.

Can I divide polynomials with remainder?

Yes. The division algorithm returns quotient Q(x) and remainder R(x) with deg R < deg D.

Which browsers are supported?

Modern evergreen browsers (desktop and mobile). No plugins required.

Polynomial Calculator

Q: Which browsers are supported?

Modern evergreen browsers (desktop and mobile). No plugins required.

A professional tool to simplify, evaluate, add, multiply, divide (quotient & remainder), differentiate, integrate, rationally factor, and plot polynomials in x. Designed for students, teachers, and engineers who need both speed and accuracy with a clear, accessible UX.

Calculator

Polynomial \(P(x)\) *

Examples: 3x^4 - 2x^2 + 5x - 7, 0.5x^3 - x + 1. Use ^ for exponent, omit the * between number and x.

Second Polynomial \(Q(x)\) (optional)

Provide when adding, multiplying, or dividing. Example: x - 2.

Operation *

Plot window [xmin,xmax,ymin,ymax]

Comma-separated. Example: -5,5,-10,10

Results

Degree—

Leading coefficient—

Canonical form—

Operation output—

Notes—

Data Source & Methodology

Authoritative data source: Classical polynomial algebra as formalized in standard textbooks such as J. Michael Steele, The Cauchy–Schwarz Master Class and undergraduate algebra references. Core rules include the polynomial division algorithm and power rules for differentiation/integration. All computations in this tool follow those definitions and properties.

“Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.”

Formulas Explained

Division Algorithm. For polynomials \(P(x)\) and \(D(x)\ne0\), there exist unique \(Q(x)\), \(R(x)\) such that

\( P(x)=D(x)\,Q(x)+R(x),\quad \deg R<\deg D. \)

Derivative (Power Rule). For \(P(x)=\sum_i a_i x^{n_i}\):

\( P'(x)=\sum_i a_i n_i x^{n_i-1}. \)

Antiderivative. For \(n\neq-1\):

\( \int a x^n\,dx=\frac{a}{n+1}x^{n+1}+C. \)

Rational Root Theorem. If \(P(x)\) has integer coefficients and a rational root \(p/q\) in lowest terms, then \(p\) divides the constant term and \(q\) divides the leading coefficient.

Glossary of Variables & Outputs

\(P(x)\): Primary polynomial (required).
\(Q(x)\): Second polynomial (optional; used for add/multiply/divide).
Degree: Highest exponent with non-zero coefficient.
Leading coefficient: Coefficient of the highest-degree term.
Canonical form: Simplified sum of terms in descending powers of \(x\).
Division output: Quotient and remainder, satisfying \(P=D\cdot Q + R\).
Factorization: Product of rational linear factors and irreducible quadratics when applicable.

How It Works: A Step-by-Step Example

Task: Divide \(P(x)=2x^3-3x^2-11x+6\) by \(D(x)=x-2\).

Enter \(P(x)\) as 2x^3 - 3x^2 - 11x + 6 and \(Q(x)\) as x - 2.
Select Divide (Quotient & Remainder) and press Calculate.
The algorithm yields \(Q(x)=2x^2+x-9\) and \(R(x)=-12\) so \(P=D\cdot Q + R\).

Frequently Asked Questions

What inputs are supported?

Expressions in one variable \(x\) with integers or decimals, plus/minus, and ^ for powers.

Is factoring exact?

Rational factors are found exactly (when present). Remaining quadratic parts are shown in standard form.

Why do I see a remainder in division?

Because \(P(x)\) may not be a multiple of \(D(x)\). By definition \(P=D\cdot Q + R\) with \(\deg R < \deg D\).

How precise is evaluation?

All arithmetic is double-precision JavaScript numbers.

How do I change the plot window?

Edit the window field (e.g., -5,5,-10,10) and press Plot.

Tool developed by Ugo Candido. Content reviewed by the CalcDomain Math Editorial Team.
Last reviewed for accuracy on: September 21, 2025.