define remainder theorem and give example

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The Remainder Theorem:

**f(x) = g(x) * q(x) + r(x)**

where g(x) is the divisor, q(x) is the quotient, and r(x) is the remainder.

The Remainder Theorem is used when dividing a polynomial expression by a linear expression of the for x - a.

For example:

Divide the polynomial x^3 - 7x - 6 by x - 4.

(x^3 - 7x - 6) / (x - 4)

x^2 + 4x + 9 with a remainder of 30.

g(x) = x - 4

q(x) = x^2 + 4x + 9

r(x) = 30

**Therefore...**

**f(x) = (x - 4) * (x^2 + 4x + 9) + 30**

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

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1 4 -9 30 The remainder 30 is the function evaluated at x = 4.

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