Polynomial Calculator
Add, subtract, multiply or divide two polynomials in x. Enter each polynomial in standard algebra syntax (e.g. 3x^2 + 2x - 5) and see the answer in descending powers along with a full term-by-term working.
What is a polynomial?
A polynomial is a sum of terms, where each term is a constant (the coefficient) times a non-negative whole-number power of the variable — usually x. The degree is the largest exponent with a nonzero coefficient. In 4x³ − 2x + 7 the terms are 4x³, −2x and 7, and the degree is 3. This calculator adds, subtracts, multiplies and long-divides them.
Polynomial operations, piece by piece
Each card explains one of the four operations the calculator performs, with the rule and a worked example matched to what you see in the step-by-step output.
Addition and subtraction — combine like terms
Terms with the same power of x are like terms; add or subtract their coefficients and keep the power. Anything with a unique power carries through unchanged. For subtraction, distribute the minus sign first: A − B = A + (−B).
Multiplication — distribute every term
Distribute every term of the first polynomial across every term of the second (the general form of FOIL). When two terms multiply, the coefficients multiply and the exponents add. Finally, combine like terms in the result.
Long division — divide, multiply, subtract, repeat
Divide the leading term of the current dividend by the leading term of the divisor to get the next quotient term. Multiply the divisor by that term, subtract from the dividend, and repeat with what's left. Stop when the remainder has a lower degree than the divisor.
Degree and standard form
The degree is the largest exponent with a nonzero coefficient — it decides the shape of the graph (line, parabola, cubic …). Answers are returned in standard form: terms ordered by descending power, with zero-coefficient terms dropped.
Features of this calculator
- Accepts natural syntax like 3x^2 + 2x - 5 with decimal or negative coefficients
- Supports addition, subtraction, multiplication and polynomial long division
- Returns the result in standard form, ordered by descending powers
- Shows term-by-term working for +, − and × and every subtraction round for ÷
- Reports the degree of the result and the remainder when dividing
Frequently asked questions
+What is the degree of a polynomial?
The largest exponent whose coefficient is not zero. The degree of a constant like 7 is 0, and the degree of the sum determines what shape its graph has (line, parabola, cubic, and so on).
+What are like terms?
Terms with the same variable raised to the same power. Only like terms can be combined by adding or subtracting their coefficients.
+How do I know when to stop polynomial long division?
Stop as soon as the current remainder has a strictly smaller degree than the divisor. What's left is the remainder; everything you built up on top is the quotient.
+Can I enter negative or decimal coefficients?
Yes — the parser accepts numbers like −3, 0.5 or 2.25 in front of any x term, and you can chain plus and minus signs naturally.