Polynomial remainder theorem

In algebra, the polynomial remainder theorem or little Bézout's theorem 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 $f(a)=0.$

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.  

See also

• Synthetic division