Let us understand the remainder theorem in polynomials with the example given below: Divide 3x3 + x2 + 2x + 5 by x + 1. Solution: From the given, Dividend = p (x) = 3x3 + x 2 + 2x + 5 Divisor = g (x) = (x + 1) Here, quotient = q (x) = 3x2 – 2x + 4 Remainder = r (x) = 1

Example 1: Find the remainder when p (x): 3x 5 − x 4 + x 3 − 4x 2 + 2 is divided by q (x): x − 1. Solution: We will substitute the zero of q (x) into the polynomial p (x) to find the remainder r. The zero of q (x) is x = 1. So by remainder theorem, r = p (1) = 3 (1) 5 - (1) 4 + (1) 3 - 4 (1) 2 + 2 = 3 - 1 + 1 - 4 + 2 = 1

It says that if you divide a polynomial, f (x), by a linear expression, x-A, the remainder will be the same as f (A). For example, the remainder when x^2 - 4x + 2 is divided by x-3 is (3)^2 - 4 (3) + 2 or -1.

Example 1: Find the remainder when you divide 3 x 2 + 2 x + 5 b y (x + 1). x − a = x + 1 Here, a = − 1. Remainder = p (a) So r = p (− 1) Substituting x = − 1, we get 3 (− 1) 2 + 2 (− 1) + 5 = 3 (1) + 2 (− 1) + 5 = 3 − 2 + 5 = 6. So, the remainder = 6.