# Polynomials

Polynomials are linear combinations of powers of x. The highest power is the order of the polynomial. For example, a cubic is a polynomial of order 3.

You may be required to manipulate polynomials including expanding brackets, simplifying expressions and factorising. There may also be questions involving Polynomial Division with the use remainder and/or factor theorem.

1. 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. $(2x-3)\left(3x^2-4x+1\right)=6x^3-8x^2+2x-9x^2+12x-3=6x^3-17x^2+14x-3$
2. Factor Theorem – Factor theorem states that if a polynomial is divisible by $x+a$ then $f(-a)=0$. It also works in the opposite direction. If $f(2)=0$, for example, then x-2 is a factor of f(x). Simplify a problem by identifying a factor of a polynomial before you use polynomial division (see below).
3. 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 $f(-a)=b$ and vice versa. The same remainder is obtained using polynomial division.
4. Polynomial Division – Polynomial Division combines algebra with the technique of long division. The idea is to identify the factor required for the left most term in each step. This factor then goes on top. For example, in Example 3, a cubic polynomial with a $2x^3$ term is being divided by a linear function with an $x$ term. The missing factor is thus $2x^2$. This is then multiplied by $x+3$ to see what remains to find.

### Example 1 – Factor Theorem

Show that $f(x)=x^5-3x^3-8$ is divisible by $x-2$.

### Example 2 – Remainder Theorem

Find the remainder when $f(x)=x^5-3x^3-8$ is divided by $x+1$.