Add, subtract, multiply and divide polynomials, or evaluate one at any value of x — with the result shown in standard form.
Enter two polynomials and choose an operation to see the result
Enter a polynomial and a value of x
This polynomial calculator is a free, all-in-one tool for single-variable polynomial arithmetic — add, subtract, multiply, and divide two polynomials in x, or evaluate a polynomial at a specific value of x. Type expressions in standard form using ^ for exponents, such as 3x^2 + 2x - 5, and the calculator returns every result formatted in standard form, highest degree first. Whether you need a quick add polynomials calculator for algebra homework, a divide polynomials calculator to check long division, or a way to evaluate a polynomial at a given x, this tool shows the operation and the resulting expression clearly.
A polynomial is an expression built from terms of the form coefficient × xᵖᵒʷᵉʳ, added or subtracted together, such as 3x² + 2x − 5. This calculator performs the four arithmetic operations (+, −, ×, ÷) between two polynomials A and B, and separately can evaluate a single polynomial P(x) at any numeric x by substitution. Division uses polynomial long division and reports both a quotient and a remainder.
Algebra and precalculus students working through polynomial arithmetic and long-division homework, engineering and physics students modeling curves or motion, computer graphics and CAD students working with polynomial-based curves, economics students building cost or revenue functions, and anyone checking hand-worked polynomial arithmetic for errors can use this tool.
Polynomials are the building blocks of algebra — they model curves, rates of change, and relationships between variables in a form that's easy to add, multiply, differentiate, and graph. Mastering polynomial add/subtract/multiply/divide operations is a prerequisite for factoring, solving polynomial equations, calculus, and virtually every higher-level algebra topic, including the quadratic and cubic equations that show up throughout STEM fields.
Engineers use polynomial expressions to model system transfer functions and signal responses. Physicists describe position under constant acceleration as a polynomial in time. Computer graphics rely on polynomial-based Bézier curves and splines to draw smooth shapes. Economists model cost, revenue, and profit as polynomial functions of quantity. Cryptographers use polynomial arithmetic over finite fields in error-correcting codes used in data storage and transmission.
^ for exponents (e.g. x^3) — a bare x is read as 1x^1, and a bare number is a constant term.How add, subtract, multiply, divide, and evaluate each work on a polynomial
Multiplying two polynomials adds their degrees. Dividing subtracts the divisor's degree from the dividend's degree to get the quotient's degree.
The coefficient on the highest-degree term controls a polynomial's end behavior and is the first number combined during multiplication.
Quotient × Divisor + Remainder must equal the original dividend exactly — the same check used for integer long division.
From typing an expression to verifying the result
Choose Add/Subtract/Multiply/Divide to operate on two polynomials, or Evaluate at x= to compute a single polynomial's value.
Type your polynomial in standard form using ^ for exponents, e.g. 3x^2 + 2x - 5.
For arithmetic mode, pick +, −, ×, or ÷ and enter Polynomial B. For Evaluate mode, enter a numeric value for x.
The calculator combines like terms, multiplies term by term, performs long division, or substitutes x, depending on the mode chosen.
The result displays in standard form, highest degree first. Division also shows the remainder in a separate box.
For division, multiply the quotient by Polynomial B and add the remainder — it should reproduce Polynomial A exactly.
All four operations plus evaluation, using A = 3x² + 2x − 5 and B = x − 1
With Polynomial A = 3x² + 2x − 5 and Polynomial B = x − 1, we'll add, subtract, multiply, and divide them, then evaluate A at x = 2.
Explanation: The division result has a remainder of exactly 0, which confirms that (x − 1) is a factor of 3x² + 2x − 5 — multiplying the quotient (3x + 5) by the divisor (x − 1) reproduces A exactly. This cross-check between multiplication and division is the fastest way to verify any polynomial arithmetic result by hand.
What each output actually represents
| Output | What It Means | Example Reading |
|---|---|---|
| Add/Subtract/Multiply result | A single polynomial in standard form, highest degree first | A × B = 3x³ − x² − 7x + 5 |
| Divide — Quotient | The polynomial that results from dividing A by B, ignoring any leftover remainder | A ÷ B quotient = 3x + 5 |
| Divide — Remainder | What's left over after division; zero means B divides A evenly (B is a factor of A) | Remainder = 0 → (x−1) is a factor of A |
| Evaluate result P(x) | The single numeric value of the polynomial at the given x | P(2) = 11 for A = 3x² + 2x − 5 |
Reading a nonzero remainder: if the remainder isn't 0, the divisor doesn't evenly divide the dividend — the true result is "quotient plus remainder ÷ divisor," similar to expressing 17 ÷ 5 as 3 remainder 2.
Manual verification: multiply the quotient back by Polynomial B and add the remainder — the result should exactly reproduce Polynomial A. For multiplication, you can also spot-check by evaluating both sides at a simple value of x, like x = 1.
Where polynomial arithmetic is genuinely useful
Check polynomial add, subtract, multiply, and long-division problems step by step.
Confirm a hand-simplified polynomial matches the calculator's standard-form output.
Explore how multiplication adds degrees and division subtracts them.
Model and compare growth rates described by polynomial expressions.
Work with polynomial expressions describing system responses and signal behavior.
Model position as a polynomial in time under constant acceleration.
Bézier curves and splines are built from polynomial functions.
Build cost, revenue, or profit functions using polynomial terms.
Polynomial arithmetic over finite fields underlies error-correcting codes.
Understand the polynomial terms used to fit curved trends to data.
Practice polynomial arithmetic questions common on algebra exams.
Double-check a textbook or hand-worked polynomial result quickly.
Reference polynomial arithmetic behind spline-based curve modeling.
What this polynomial calculator does well, and where manual judgment is still needed
^ for exponents rather than accepting free-form expression syntaxClassifying expressions by their number of terms
| Name | Number of Terms | Example |
|---|---|---|
| Monomial | 1 term | 5x² |
| Binomial | 2 terms | x + 3 |
| Trinomial | 3 terms | x² + 2x − 5 |
| Polynomial (general term) | 1 or more terms | 3x³ − x² − 7x + 5 |
^ symbol for exponents, causing the parser to misread the expressionSummary: This polynomial calculator gives you an instant, free way to add, subtract, multiply, and divide polynomials in x, and to evaluate a polynomial at any value — all shown in standard form with quotient and remainder separated. Pair it with the Factoring Calculator and Quadratic Equation Solver for a fuller picture of polynomial algebra.
Common questions about polynomial arithmetic
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