Polynomial long division calculator

How it works

Dividing a polynomial $p(x)$ by $d(x)$ gives a quotient $q(x)$ and a remainder $r(x)$ with $p = q \cdot d + r$ and $\deg r < \deg d$, just like whole number division. Long division finds them one term at a time: divide the leading term of what is left by the leading term of $d$, write the result in the quotient, multiply it by $d$ and subtract. Each round lowers the degree, and the work stops when the degree of what is left is smaller than that of $d$.

When the divisor is $x - c$, synthetic division (Horner’s scheme) does the same on the coefficients alone: bring down the first one, then repeatedly multiply by $c$ and add the next. The last number is the remainder, and it equals $p(c)$; so $x - c$ divides $p$ exactly when $c$ is a root.

Polynomial division over the fractions is the base of the Euclidean algorithm for polynomials. In cryptography the same division runs over finite fields: AES computes in $\mathbb{F}_{2^8}$ modulo $x^8 + x^4 + x^3 + x + 1$, and CRC checksums are remainders of polynomial division over $\mathbb{F}_2$.