Identity for fractions

< 1 >

The identity for fractions is written as

Dividend ≡ Divisor × Quotient + Remainder

There is always a remainder, but that may possibly be the value zero. For a function in the form

the remainder theorem applies. If you divide f (x) by (x – a), then the remainder is f (a). The operation is

If f (x) is of degree n, then q (x) is a form of degree (n – 1), while the remainder R no longer contains x and is only a number. Therefore

This is an identity that applies for each value of x, so also for x = a. Substitution gives

So you do not need to perform a division to obtain the remainder.

A simple fraction gives

With a long division we calculate the remainder of

Now we write

and because this identity is also valid for x = 2, you get