Polynome Remainder

This assumes:

• Polynomials
• [Polynome Division](Polynome Division)

This is a trick one can use to determine the remainder or the polynome input (or the other way). If the polynome $P(x)$ is divided by $(x - x_{0})$ then the remainder is the same as calculating $P(x_{0})$. In this case $x_{0}$ is the zero-point of the divisor in the division.

Example

Our polynome:

$P(x) = x^2 + x - 1$

Our divisor:

$(x - 2)$

It’s zero-point:

$(x - 2 = 0) | + 2 = (x = 2)$

The hack:

Here I have made use of a fictive function returning the remainder from the division:

$r = Remainder((x^2 + x + 1) \div (x - 2)) = P(2)$

Demonstration:

We will now calculate the remainder without performing the division:

1. $r = P(x) = x^2 + x - 1$
2. $r = P(2) = 2^2 + 2 - 1$
3. $r = P(2) = 4 + 2 - 1$
4. $r = P(2) = 6 - 1$
5. $r = P(2) = 5$
6. $r = 5$