Can I have an explanation of how the remainder function works?  

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The remainder function is an easy way of determining whether a term ( x - a) is a factor of a polynomial P(x).

Whenever P(x) is divided by (x - a), the value of P(a) is the remainder that P(x) divided by (x - a) would yield. To find out if (x - a) is a factor of P(x), we substitute x with a. P(a) = 0 indicates that P(x) is divisible by (x - a)

The proof of the remainder theorem is:

Take a polynomial P(x); when it is divided by (x – a) let the quotient be denoted by q and the remainder by r. This gives: P(x) = q*(x – a) + r. When we substitute x with a, we have P(a) = q*(a – a) + r. Or r = P(a).

This proves that P(a) is the remainder when P(x) is divided by (x – a).

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