define remainder theorem and give example

The Remainder Theorem:

Given
a polynomial f(x), and a divisor (x-a).  If r is the remainder of f(x) divided by (x-a)
then f(a) = r.

Proof:

Let
f(x)/(x-a) = g(x)*q(x) + r/(x-a)   Now multiply everything by (x-a) and we
get

f(x) = g(x)*q(x)*(x-a) + r   Now evaluate at
x=a

f(a) = g(a)*q(a)*(a-a) + r  Now note (a-a) = 0 and
anything multiplied by 0 is 0

so f(a) = g(a)*q(a)*(0) + r =
0 + r = r

So f(a) = r

In the
above example f(4) = 30

f(4) = 4^3 - 7(4) - 6 = 64 - 28 - 6
= 36 - 6 = 30.

The remainder theorem is used in using
synthetic division to evaluate a polynomial because synthetic division is simplier,
generally produces smaller numbers, and less error prone than
evaluation.

4) 1 0   -7  -6
4  16
36
-----------------
1 4   -9 30   The remainder 30 is the function
evaluated at x = 4.