This document discusses the remainder theorem for polynomials. The remainder theorem states that if a polynomial P(x) is divided by a linear divisor (x - a), the remainder is equal to P(a). The document provides examples of using the remainder theorem to find the remainder when dividing polynomials by linear expressions like (x + 1) or (x - a). It explains that the remainder theorem provides a simpler method than long division for calculating the remainder.

1. Polynomials and Partial Fractions Objectives In this lesson, you will learn how to find the remainder when a polynomial is divided by a linear divisor, by an elegant method - using the remainder theorem . 4.4 Remainder Theorem

2. In the previous lesson, we saw that: We will generalise this result as the remainder theorem. Polynomials and Partial Fractions Then: And if x = 1:

3. If a polynomial P( x ) is divided by a linear divisor ( x – a ), the remainder is P( a ). The remainder theorem is a much simpler and more elegant way of finding the remainder compared to long division. Polynomials and Partial Fractions Then: And if x = a : Let Q( x ) be the quotient and R be the remainder.

4. Let x = – 1. The remainder is 6. Find the remainder when . Polynomials and Partial Fractions By the remainder theorem, when P( x ) is divided by x + 1, Substitute for x in P( x ). Example

5. Let x = – 2 a. Polynomials and Partial Fractions Find the possible values of a. Substitute for x in P( x ). By the remainder theorem, when P( x ) is divided by x + 2 a, R = 32. There are two possible values of a . Example

6. Let x = 1. Polynomials and Partial Fractions Substitute for x in P( x ). By the remainder theorem, when P( x ) is divided by x – 1, Equating remainders Shown By the remainder theorem, when P( x ) is divided by x + 1, Substitute for x in P( x ). Let x = – 1. Example .