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Polynomial Remainder Theorem

Theorem: Remainder of division of a polynomial f(x) by a linear factor (x-a) is f(a)

Prerequisites:
Euclidean Polynomials Division (proof)

Proof:
Any polynomial can be written according to the Euclidean polynomial division as:

\qquad\quad f(x) = q(x)g(x) + r(x)\qquad\text{where }\;\; deg\big(r(x)\big) < deg\big(g(x)\big)\qquad\qquad\ldots (1)

Here q(x), g(x) and r(x) are the quotient, divisor and remainder respectively and deg(.) represents the degree of the respective polynomial.

Since g(x) = (x-a), therefore  deg\big(g(x)\big) = 1

Hence, deg\big(r(x)\big) = 0
\Rightarrow \quad\;\; r(x) = r = constant
\therefore \qquad\! r(x) is independent of x.

Putting x=a and g(x) = (x-a) in equation (1),
\qquad\quad f(a) = q(a)(a-a) + r\\ \Rightarrow\;\quad\; f(a) = r

Hence the result


Recommended:
Factor Theorem
Factor representation of polynomials

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