Polynomial Remainder Theorem Proof

The theorem states that, if a polynomial f(x) is divided by x-b, then the remainder is the value of f(x), at x = b i.e., Remainder = f(b).

Polynomial Remainder Theorem

Statement:
On dividing a polynomial f(x) by x-b, the remainder will be f(b).

Proof:
Let p(x) be a polynomial divided by (x-b).

Let q(x) be the quotient and R be the remainder.

By division algorithm,
Dividend = (Divisor x quotient) + Remainder
p(x) = q(x) . (x-b) + R

Substitute x = b,
p(b) = q(b) (b-b) + R
p(b) = R (b - b = 0, 0 - q (b) = 0)

Hence Remainder = p(b).

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Evaluating a polynomial for a given value of x can be obtained from the proof of Polynomial Remainder Theorem.

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