Polynomial long division

Polynomial long division is a method used to divide one polynomial by another polynomial, similar to the long division process used with numbers. It involves a systematic way of dividing polynomials, which results in a quotient and, in some cases, a remainder.

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Polynomial long division by

Codex  0 Created 2026-10-14 Updated 2026-10-18

Start with a provisional polynomial quotient

q = 0

h = f

. If

h

has polynomial degree

n

at least the divisor's polynomial degree

m

, with leading coefficients

a

and

b

, subtract

t g

, where

t = \frac{a}{b} X^{n - m}, q ⟵ q + t, h ⟵ h - t g .

The identity

f = g q + h

survives each step, while the leading term cancels and the polynomial degree of nonzero

h

strictly decreases. Stop when

h = 0

or

de g h < m

.

For uniqueness over a field, subtract two proposed decompositions:

g (q_{1} - q_{2}) = r_{2} - r_{1} .

If

q_{1} \neq = q_{2}

, the left side has polynomial degree at least

de g g

, while the right side is zero or has smaller polynomial degree. Hence

q_{1} = q_{2}

and then

r_{1} = r_{2}

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