Solved Examples Find the remainder when f (x) = 5x4 – 2x3 + 7x2 – 10 is divided by (x – 2) Solution: Here, f (x) = 5x 4 – 2x 3 + 7x 2 – 10 By setting x – 2 = 0 ⇒ x = 2 Thus, the remainder r (x) = f (2) = 5 (2) 4 – 2 (2) 3 + 7 (2) 2 – 10 = 82 Use the remainder theorem to find whether x = -4 is a zero of f (x) = x3 – 11x2 + 14x – 22

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

Let’s take a look at the application of the remainder theorem with the help of an example. Example 1: Find the remainder when t 3 – 2t 2 + t + 1 is divided by t – 1. Solution: Here, p (x) = t 3 – 2t 2 + t + 1, and the zero of t – 1 is 1. ∴ p (1) = (1) 3 – 2 (1) 2 + 1 + 1= 1