Polynomial remainder theorem

Not to be confused with Bézout's theorem.

In algebra, the polynomial remainder theorem or little Bézout's theorem is an application of polynomial long division. It states that the remainder of a polynomial $f(x)\,$ divided by a linear divisor $x-a\,$ is equal to $f(a) \,.$

Example

Let $f(x) = x^3 - 12x^2 - 42\,$. Polynomial division of $f(x)\,$ by $x-3\,$ gives the quotient $x^2 - 9x - 27\,$ and the remainder $-123\,$. Therefore, $f(3)=-123\,$.

Proof

The polynomial remainder theorem follows from the definition of polynomial long division; denoting the divisor, quotient and remainder by, respectively, $g(x)\,$, $q(x)\,$, and $r(x)\,$, polynomial long division gives a solution of the equation

$f(x)=q(x)g(x) + r(x)\,,$

where the degree of $r(x)\,$ is less than that of $g(x)\,$.

If we take $g(x) = x-a\,$ as the divisor, giving the degree of $r(x)\,$ as 0, i.e. $r(x) = r\,$:

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

Setting $x=a \!\,$ we obtain:

$f(a)=r\,.$

Applications

The polynomial remainder theorem may be used to evaluate $f(a)\,$ by calculating the remainder, r. Although polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem.

The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.