Chemistry - Polynomial remainder theorem

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

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

$\frac{f(x)}{g(x)}=q(x) + \frac{r}{g(x)}$

Set the divisor $g (x)$ to the linear divisor $x - a$ :

$\frac{f(x)}{x-a}=q(x) + \frac{r}{x-a}$

Solve for $f (x)$ :

$\frac{}{}f(x)=q(x)(x-a) + r$

Set $x = a$ :

$\frac{}{}f(a)=r$

The polynomial remainder theorem may be used to evaluate $f (a)$ by calculating the remainder $r$ . 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 $f (x) / (x - 3)$ is 123 so $f (3) = 123$ .

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.

Categories: Polynomials | Theorems