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Polynomial Division Calculator

Divide one single-variable polynomial by another and get the exact quotient, exact remainder, and the verified identity in one place.

Parsed dividend
Parsed divisor
Examples

x^3 - 1 divided by x - 1 gives quotient x^2 + x + 1 and remainder 0.

Quotient
x^{2} + x + 1
Remainder
0
Verified identity
\left(x^{3} - 1\right) = \left(x - 1\right) \cdot \left(x^{2} + x + 1\right) + \left(0\right)
Long-division trail
x^{3} \div x = x^{2}, x^{2} \cdot \left(x - 1\right) = x^{3} - x^{2}, \left(x^{3} - 1\right) - \left(x^{3} - x^{2}\right) = x^{2} - 1, x^{2} \div x = x, x \cdot \left(x - 1\right) = x^{2} - x, \left(x^{2} - 1\right) - \left(x^{2} - x\right) = x - 1, x \div x = 1, 1 \cdot \left(x - 1\right) = x - 1, \left(x - 1\right) - \left(x - 1\right) = 0, Q = x^{2} + x + 1,\; R = 0

Remainder is exactly 0, so x - 1 is a factor of x^3 - 1.

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Examples

How It Works

Formula

P(x)=D(x)⋅Q(x)+R(x)P(x) = D(x) \cdot Q(x) + R(x)

deg⁡(R)<deg⁡(D)\deg(R) < \deg(D)

Variables

P(x)P(x)

Dividend polynomial

D(x)D(x)

Divisor polynomial

Q(x)Q(x)

Quotient polynomial

R(x)R(x)

Remainder polynomial

The calculator parses both expressions as exact single-variable polynomials, divides leading terms, subtracts the matching partial product, and repeats until the remainder degree is lower than the divisor degree. Coefficients stay symbolic, so a result like 1/2x^2 - 1/2x + 1 remains exact instead of being turned into decimals.

At each pass the leading term of the current remainder is divided by the leading term of the divisor. That quotient term is multiplied back through the divisor, then subtracted from the current remainder. The process stops only when the remaining polynomial has smaller degree than the divisor, which is the standard long-division rule used in algebra classes.

Frequently Asked Questions

01What does this calculator return?
It returns the exact quotient polynomial, the exact remainder polynomial, and the identity dividend = divisor × quotient + remainder.
02When is the divisor a factor?
Only when the remainder is exactly 0. In that case the dividend is divisible by the divisor with no leftover term.
03What input style is supported?
Enter a single-variable polynomial with standard algebra notation, such as x^4 - 3x + 2, x^2/2 - 1, or (x + 1)^2 - x. The calculator normalizes missing terms automatically.
04What is not supported?
This tool is not a full CAS. It does not accept functions like sin(x), expressions with negative exponents, or divisions by variable expressions such as 1/x.
05Why show the identity below the answer?
It gives a quick correctness check. If you expand divisor × quotient and add the remainder, you should recover the original dividend.

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