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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.