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For each division a remainder can be calculated by

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

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


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