## Manipulating Polynomials

You may be required to manipulate polynomials including expanding brackets, simplifying expressions, factorising and polynomial division possibly with the use of remainder and factor theorem.

### Expand and Simplify

Expand and simpify – this works in the same way as expanding double brackets but you should end up with 6 terms before simplification.

### Factor Theorem

**Factor theorem states that if a polynomial is divisible by then and vice versa.**

*Example* – show that is divisible by x-2. The easiest way to do this is to use factor theorem which states that f(x) is divisible by x-2 if f(2)=0. , hence x-2 is a factor of f(x) and f(x) is divisible by x-2.

### Remainder Theorem

**Remainder theorem states that if a polynomial is not divisible by then dividing by will result in a remainder in which case .**

Example – find the remainder when is divided by x+1. The simplest way to do this is to use remainder theorem which states that the remainder when f(x) is divided by x+1 is f(-1). and so the remainder when f(x) is divided by x+1 is -6.