Algebraic Division,Factor Theorem and Remainder Theorem

•  Edexcel, Module - C2, Chapter - Algebra and Functions
• AQA, Module - C1, Chapter - Algebra
• OCR, Module - C2, Chapter - Algebra

First chapter of C2 is Algebra and Functions again. We firstly start off with Algebraic Division, this is basically the long division we've done with numbers, except now we have polynomials. Before proceeding, it's good to know what a polynomial is.. and how to factorise certain polynomials (cubics, quadratics).. if you've forgotten. look into my c1 posts.

We are required to divide polynomials by (x-a ) or (x+ a), where a is some integer. This division process can either result in (a being a factor, which will mean after the division is complete, the remainder will be 0, and we will obtain a quotient (which is usually a degree less of the polynomial being divided). OR the division can result in a remainder, in which a is not a factor.

I will show you the actual process and examples in my image notes, it is hard to write it on here. The process is always the same. A couple of common errors :

* Say we have x^3 + 3x + 2. and we have to divide it by some factor (x-a).. some students may not be able to continue after some point, it is simply because we have to include the x^2 term in the division, so we rewrite the polynomial as :  x^3 + 0x^2 + 3x + 2 divided by (x-a).  [Always include terms that are not present by writing 0x^a (a being whatever power).. this will line up the columns]

General Teminology

Factor - Something that is completely divisible by another thing, (no remainder) x+4 is a factor of x^2 + 8x + 16, 2 is a factor of 10, 10 is a factor of 150.. etc.

Remainder - The value obtained / left over when one polynomial is divided by another, where it is not a factor. The Remainder Theorem can be used to find the remainder, instead of long division.

Divisor - The polynomial you are dividing by. e.g x^2 - x + 1 divided by (x - 2 < divisor)

Quotient - The expression obtained as a result of dividing polynomials. e.g. ( in numbers 8 /4 = 2 - quotient)

Quotient * Divisor = Dividend (original expression)
or
(Quotient*Divisor) + Remainder = Dividend (original expression)


Factor Theorem

The Factor Theorem states, that if f(a)=0, then (x-a) is a factor. This is another way of checking if some expression is completely divisible by another.

Remainder Theorem
Same as the factor theorem, when you divide a polynomial by (x-a), the remainder will be f(a).

Notes of Algebraic Division 2

Notes for Algebraic Divsion 1