Polynomial Calculator

How to Use the Polynomial Calculator

Enter a polynomial

Type it in natural notation, like x^2 + 2x + 5 or 2x^3 - 3x + 1. Use ^ for powers and the usual + and − signs.

Choose an operation

Evaluate it at a value of x, or add, subtract, multiply, or divide it by a second polynomial.

Enter the second polynomial or x

For arithmetic, type the second polynomial. For evaluation, type the value of x to substitute.

Read the result

You get the resulting polynomial in tidy form, with its degree and leading coefficient — and for division, the quotient and remainder separately.

Working With Polynomials

Algebra's Workhorse

A polynomial is an expression built from a variable and numbers using only addition, subtraction, multiplication, and whole-number powers — things like x² + 2x + 5 or 3x⁴ − x + 7. They are the central objects of algebra for a simple reason: they are flexible enough to model an enormous range of relationships, yet simple enough to manipulate with clean rules. Adding and subtracting polynomials just means combining like terms — the x² terms together, the x terms together, the constants together. Multiplying them means distributing every term of one across every term of the other (the generalisation of the "FOIL" rule). This calculator does all of that exactly, keeping track of every coefficient, so you can check homework, expand a product, or combine expressions without the slips that creep in when doing it by hand.

The genuinely tricky operation — and the one students most often search for — is polynomial long division. Dividing one polynomial by another works just like long division of numbers: you see how many times the leading term of the divisor goes into the leading term of the dividend, multiply back, subtract, and repeat, producing a quotient and a remainder. It is the polynomial echo of the long division you learned with numbers, and it is the gateway to factoring, partial fractions, and the remainder and factor theorems. This calculator shows the quotient and remainder cleanly, and it can also evaluate a polynomial at any value using Horner's method — an efficient way of nesting the multiplications that both computes the value and underlies fast, accurate polynomial evaluation in computing. Evaluation is more than a convenience: by the remainder theorem, the value of a polynomial at x = a is exactly the remainder when you divide by (x − a), which is why evaluation and division are two sides of one idea.

"Polynomials are algebra's workhorse: flexible enough to model the world, simple enough to add, multiply, and divide by clean rules. Long division is where the real machinery shows."

Why Polynomials Matter

Beyond the classroom, polynomials are everywhere in applied mathematics. They approximate complicated functions (a Taylor series is a polynomial), they define the smooth curves in computer graphics and fonts (Bézier curves are polynomials), they appear in error-correcting codes that keep your data intact, and they are the basis of regression models that fit trends to data. The operations this calculator performs — combining, expanding, dividing, and evaluating — are the elementary moves from which all of that is built. Because it runs entirely in your browser and parses ordinary written notation, it is an instant, private way to manipulate polynomials accurately: paste in the expression from your textbook or worksheet, pick the operation, and get a clean, correct result with its degree and leading coefficient, every time.

10 Facts About Polynomials

01

A polynomial uses only +, −, ×, and whole-number powers of a variable.

02

The degree is the highest power; the leading coefficient sits in front of it.

03

To add polynomials, you combine like terms.

04

To multiply, you distribute every term across every other — the general FOIL.

05

Long division of polynomials mirrors long division of numbers.

06

The remainder theorem: P(a) equals the remainder when dividing by (x − a).

07

Horner's method evaluates a polynomial with the fewest multiplications.

08

A Taylor series approximates any smooth function with a polynomial.

09

Bézier curves in graphics and fonts are polynomials.

10

This tool keeps every coefficient exact as it computes.

Frequently Asked Questions

Type it in ordinary notation using ^ for powers — for example x^2 + 2x + 5 or 2x^3 - 3x + 1. The calculator understands plus and minus signs, implicit coefficients (x means 1x), and constant terms. You do not need to write the terms in any particular order.
It works just like long division of numbers: divide the leading terms, multiply the divisor by that quotient term, subtract, and repeat with what is left, until the remainder has a lower degree than the divisor. The calculator gives the quotient and remainder separately, which is exactly the form students are usually asked for.
It means substituting a number for x and computing the result — for example, x² + 2x + 5 at x = 3 is 9 + 6 + 5 = 20. The calculator uses Horner's method, which nests the multiplications efficiently. By the remainder theorem, this value also equals the remainder when dividing by (x − 3).
The degree is the highest power of x that appears, and the leading coefficient is the number multiplying that highest power. For 2x³ − 3x + 1, the degree is 3 and the leading coefficient is 2. The calculator reports both for every result, since they describe the polynomial's overall shape and behaviour.
Yes — choose Multiply and enter both polynomials. The calculator distributes every term of the first across every term of the second and combines like terms, giving the fully expanded product. This is the general version of the FOIL rule and works for any degrees.
It says the value of a polynomial P at x = a is exactly the remainder when you divide P by (x − a). So if P(a) = 0, then (x − a) divides P evenly — that is the factor theorem, the foundation of factoring. You can see this connection by comparing the Evaluate and Divide modes.
This tool focuses on arithmetic and evaluation. For factoring into linear and quadratic pieces and finding roots, use the dedicated Factoring Calculator, which is built on the same engine and pairs naturally with this one.
Yes. Everything is computed in your browser — nothing is uploaded, stored, or logged — and it works offline once loaded.
Completely free, with no account, sign-up, or limit. It runs entirely in your browser and collects no data. Use it as often as you like.
No. You can enter the terms in any order — the calculator parses them and presents the result in standard form, from the highest power down. Repeated powers are combined automatically.

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