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In algebra, the polynomial remainder theorem or little Bézout's theorem[1] is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial $f(x)$ by a linear polynomial $x-a$ is equal to $f(a) .$ In particular, $x-a$ is a divisor of $f(x)$ if and only if $f(a)=0.$[2]

## Proof

The polynomial remainder theorem follows from the definition of Euclidean division, which, given two polynomials f(x) (the dividend) and g(x) (the divisor), asserts the existence and the uniqueness of a quotient q(x) and a remainder r(x) such that

$f(x)=q(x)g(x) + r(x)\quad \text{and}\quad r(x) =0 \; \text{or}\; \deg(r)<\deg(g)\,.$

If we take $g(x) = x-a$ as the divisor, either r = 0 or its degree is zero; in both cases, r is a constant that is independent of x; that is

$f(x)=q(x)(x-a) + r\,.$

Setting $x=a$ in this formula, we obtain:

$f(a)=r\,.$

A slightly different proof, which may appear to some people as more elementary, starts with an observation that $f(x)-f(a)$ is a linear combination of the terms of the form $x^k-a^k$, each of which is divisible by $x-a$ since $x^k-a^k=(x-a)(x^{k-1}+x^{k-2}a+\dots+xa^{k-2}+a^{k-1})$.

The polynominal remainder theorem can be used as a shorter means to obtain a factor compared to polynomial long division.