## 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 $(2x-3)\left(3x^2-4x+1\right)$ – this works in the same way as expanding double brackets but you should end up with 6 terms before simplification. $(2x-3)\left(3x^2-4x+1\right)=6x^3-8x^2+2x-9x^2+12x-3=6x^3-17x^2+14x-3$

### Factor Theorem

Factor theorem states that if a polynomial is divisible by $x+a$ then $f(-a)=0$ and vice versa.

Example – show that $f(x)=x^5-3x^3-8$ 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. $f(2)=2^5-3(2)^3-8=32-24-8=0$, 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 $x+a$ then dividing by $x+a$ will result in a remainder $b$ in which case $f(-a)=b$.
Example – find the remainder when $f(x)=x^5-3x^3-8$ 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). $f(-1)=(-1)^5-3(-1)^3-8=-1+3-8=-6$ and so the remainder when f(x) is divided by x+1 is -6.