### Remainder

For each division a remainder can be calculated by

Dividend ≡ Divisor × Quotient + Remainder

##### Explanation

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

or written as a long division

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.

##### Example 1

A simple fraction gives

##### Example 2

With a long division we calculate the remainder of

Now we write

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