Polynomial Calculator

Polynomial (in x)

Operation

Canonical form—

Degree—

Result—

Polynomial Calculator – Free Online Solver with Steps

Our free Polynomial Calculator helps you perform operations with polynomials instantly and step-by-step. Whether you need to add, subtract, multiply, divide, factor, or expand polynomials, this tool makes algebra easier. It also supports synthetic division, long division, and finding roots of polynomial equations.

What Is a Polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients, combined with addition, subtraction, and multiplication, where exponents are whole numbers.

Examples:

Linear: 2x + 5
Quadratic: x² - 3x + 2
Cubic: x³ + 4x² - x - 6
Higher degree: 5x⁴ - 2x³ + 7

Features of the Polynomial Calculator

Polynomial Operations: add, subtract, multiply, and expand expressions.
✅ Polynomial Division: perform synthetic division and long division with step-by-step results.
✅ Root Finder: solve polynomial equations for real and complex roots.
✅ Factorisation: factor polynomials into simpler terms.
✅ Taylor Polynomial Generator: create approximations of functions.
✅ Graphing (optional): plot polynomials to visualise behaviour.

How to Use the Polynomial Solver

Enter your polynomial expression (e.g., x³ + 2x² − 5x − 6).

Choose an operation:

Simplify
Expand
Factorise
Divide (synthetic or long division)
Solve equation P(x) = 0

Click Calculate. See the solution with detailed steps.

Synthetic Division Example

Divide 2x³ + 3x² − 5x + 6 by (x − 2):

Set up coefficients: 2, 3, −5, 6.
Write divisor root: x − 2 → r = 2.
Perform synthetic division:

2 |  2    3    -5    6
    |       4    14   18
    |  2    7     9   24

Quotient: 2x² + 7x + 9
Remainder: 24

So: (2x³ + 3x² − 5x + 6) ÷ (x − 2) = 2x² + 7x + 9 + 24/(x − 2)

Polynomial Equation Solver

To solve x³ − 6x² + 11x − 6 = 0:

Factorise → (x − 1)(x − 2)(x − 3)

Roots: x = 1, 2, 3

Taylor Polynomial Formula

The Taylor polynomial of degree n for a function f(x) around a is:

P_n(x) = f(a) + f′(a)(x − a) + ^f″(a)/_2!(x − a)² + … + ^{f⁽ⁿ⁾(a)}/_n!(x − a)ⁿ

Our tool can generate Taylor expansions for functions like sin(x), eˣ, and more.

FAQ – Polynomial Calculator

Q: Can this calculator do synthetic division?
Yes ✅ — enter your polynomial and divisor, and it shows every step of the synthetic division.

Q: Can it solve polynomial equations?
Yes — it finds exact roots (if rational) or approximates irrational/complex roots.

Q: Does it handle factoring?
Yes — it factors polynomials into linear or quadratic factors where possible.

Q: Can it expand or simplify expressions?
Yes — you can expand (x + 2)(x - 3) into x² - x - 6.

Q: Can it generate Taylor polynomials?
Yes — choose the function and degree, and the tool gives the Taylor polynomial formula.