Polynomial remainder theorem

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

This can be demonstrated by the definition of polynomial long division:

<math>\frac{f(x)}{g(x)}=q(x) + \frac{r}{g(x)}<math>

Set the divisor <math>g(x)<math> to the linear divisor <math>x-a<math>:

<math>\frac{f(x)}{x-a}=q(x) + \frac{r}{x-a}<math>

Solve for <math>f(x)<math>:

<math>\frac{}{}f(x)=q(x)(x-a) + r<math>

Set <math>x=a<math>:

<math>\frac{}{}f(a)=r<math>

The polynomial remainder theorem may be used to evaluate <math>f(a)<math> by calculating the remainder <math>r<math>. While polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. In the example in the polynomial long division article, the remainder of <math>f(x)/(x-3)<math> is 123 so <math>f(3)=123<math>.

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