Polynomial Long Division¶

Definition¶

Polynomial long division is the procedure that computes the quotient and remainder in the division algorithm for polynomials: for nonzero polynomials $a(x),b(x)\in\mathbb{F}[x]$ $F [x]$ , the polynomials with coefficients in a field $\mathbb{F}$ , there are unique polynomials $q(x)$ and $r(x)$ such that

a(x)=q(x)b(x)+r(x),

with $r(x)=0$ or $\deg(r)<\deg(b)$ $de g (b)$ .

Interpretation¶

Polynomial long division answers the question how often $b(x)$ fits into $a(x)$ and what is left over, just as integer division with remainder writes $17=3\cdot5+2$ with $0\le2<5$ $5$ . For polynomials, the size of the remainder is measured by its degree instead of its value.

Each step looks only at leading terms. If what is left of $a(x)$ has leading term $c\,x^m$ and $b(x)$ has leading term $d\,x^k$ with $m\ge k$ $k$ , the next term of the quotient is $\tfrac{c}{d}x^{m-k}$ . Subtracting $\tfrac{c}{d}x^{m-k}b(x)$ cancels the leading term and lowers the degree. The process stops as soon as what is left is $0$ or has degree smaller than $\deg(b)$ , and that is $r(x)$ . This is where the field matters: the step divides by the leading coefficient $d$ , which needs its inverse $d^{-1}$ . With integer coefficients, dividing $x$ by $2x$ would already need the quotient $\tfrac{1}{2}$ .

The most frequent case is division by $x-\lambda$ $λ$ . The remainder is then a constant, and substituting $x=\lambda$ in $a(x)=q(x)(x-\lambda)+r$ shows that it equals $a(\lambda)$ . So $\lambda$ is a root of $a$ if and only if the division leaves remainder $0$ . Repeated division is also the engine of the Euclidean algorithm for the greatest common divisor of two polynomials.

Key Results¶

The quotient and remainder are uniquely determined.
Polynomial division gives a quotient and remainder with the remainder of smaller degree.

Examples¶

Dividing $x^2-1$ by $x-1$ gives quotient $x+1$ and remainder $0$ .
Dividing $x^3+2x+1$ by $x^2+1$ gives quotient $x$ and remainder $x+1$ $1$ : $x^3+2x+1=x\,(x^2+1)+(x+1)$ $1)$ , with $\deg(x+1)=1<2$ $2$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: polynomial division, long division of polynomials Symbols: element of, lambda

Last updated: 20 October 2026 03:02