Polynomial remainder theorem

The polynomial remainder theorem in algebra is an application of polynomial long division. It states that the remainder, $r,$, of a polynomial, $f(x),$, divided by a linear divisor, $x-a,$, is equal to $f(a)\; ,.$

This 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::$f(a)=r,.$

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.

Example

Let $f(x)\; =\; x^3\; -\; 12x^2\; -\; 42,$.

Polynomial division by $x-3,$ gives the quotient

$x^2\; -\; 9x\; -\; 27,$ and the remainder $-123,$.

Therefore, $f(3)=-123,$.