Remainder Theorem

Remainder Theorem

Let p(x) be a polynomial in x of degree greater than or equal to 1 and 'a' be any real number. If p(x) is divided by (x – a), then the remainder is p(a).

If p(x) is divided by (x – a), then the remainder is r(x) and the quotient is q(x). Thus, p(x) = (x - a) q(x) + r(x).

The degree of r(x) is always less than the degree of (x - a). Since the degree of (x - a) is one, the degree of r(x) is zero i.e. r(x) is a constant.

So, for every value of x, r(x) = r. Therefore, p(x) = (x – a) q(x) + r

Based on the remainder theorem:

• If a polynomial p(x) is divided by (x + a), then the remainder is p(– a).
• If a polynomial p(x) is divided by (ax – b), then the remainder is p($\frac{\text{b}}{\text{a}}$).
• If a polynomial p(x) is divided by (ax + b), then the remainder is p($\frac{\text{-b}}{\text{a}}$).
• If a polynomial p(x) is divided by (b – ax), then the remainder is p($\frac{\text{b}}{\text{a}}$).