Introduction

A Polynomial is an expression that comprises of more than 1 positive integer powers of a variable (in addition to one positive constant term where the power is 0)
Integer - a positive, negative or 0 whole number
The order or degree​ of a polynomial is the highest power of the variable.

Background Knowledge​

• Expanding and factorising linear and quadratic expressions
• Solving linear and quadratics
• Algebraic Manipulation
• Function Notation

Addition, Subtraction and Multiplication

Addition and Subtraction of polynomials is simply collecting like terms. This can also be done in columns to make it easier.

The multiplication of polynomials is simply expanding brackets where each term in one is multiplied by each term in the others and then like terms are collected.

Division

Polynomials can also be divided. When this happens, there is a polynomial, a divisor, a quotient and sometimes a remainder.
The Polynomial: P(x)​ is what is being divided
The Divisor: D(x) is what the polynomial is being divided by
The Quotient: Q(x) is the result of the division
The Remainder: R(x) (if there is one) is what is left after the division

• Long Division
• Comparing Coefficients

<

>

If there is a remainder, or you don't know whether there will be a remainder or not, you should use long division...

• Set up as shown in black below
• Divide the highest power term in the polynomial by the highest power term in the divisor and write the result above
• Then multiply the term of the quotient you just wrote by the divisor and write it below the term you divided in the polynomial and underneath the following term in such a way that the powers line up
• Subtract that from the part of the polynomial it is under, writing the result underneath
• Bring down the next term
• Divide the result of the earlier subtraction by the highest power in the divisor and write the result above
• Again, multiply the term of the quotient you just wrote by the divisor and write it below the term you just divided (the result of the previous subtraction) so the powers line up
• Subtract that from the terms you wrote it below and write the result underneath.
• Bring down the next term and continue the cycle till all terms have been divided
• There will either be a 0 at the end or a remainder

If you know there is no remainder, it is possible to compare coefficients to divide. A coefficient is the number before the variable(s) in a term and you can compare the coefficients of the polynomial to the coefficients of the quadratic, using the factor. Here are two different ways of looking at it. 

You may then be asked to factorise the polynomial completely when you have been given one factor. To do this, find the quotient and then factorise it if need be.

Factor Theorem and Remainder Theorem

Polynomials can also be represented by functions (as shown earlier when naming the parts of a polynomial division): $$ f(x) = x^3 + 2x^2 + 3x +4 $$
This means, if I put a value of x into f(x), I will get the result of substituting x into the polynomial.
Examples:

• f(1) = 10
• f(2) = 26
• f(0) = 4

Factor theorem states that If (x-a) is a factor of f(x), f(a) = 0
This means...

• If (x+7) is a factor of f(x), f(-7) = 0
• If f(-3) = 0, (x+3) is a factor of f(x)
• If (x-1) is a factor of f(x), f(1) = 0
• If f(117) = 0, (x-117) is a factor of f(x)

Remainder theorem states that P(x) = Q(x)*D(x) + R(x)
As for a polynomial with no remainder is as follows: P(x) = Q(x)*D(x) and factor theorem applies, when there is a remainder...
When f(x) = 0, x is the remainder