Is Division algorithm valid when divisor is zero?

Question

Can division algorithm be valid when the divisor is zero

Suppose I’ve polynomial $g(x)=x-1$, $p(x)$, $q(x)$, $r(x)$ and when I divide $p(x)$ by $g(x)$ i get $q(x)$ as quotient and $r(x)$ as the remainder. According to division algorithm

$$p(x)= q(x)(x-1)+r(x)$$

Now my country’s textbook proves the remainder theorem by substituting $x=1$ in division algorithm but I want to ask that can we use the division algorithm even when the divisor is zero. I think we can’t since division by zero is not defined and thus division algorithm for the same will also be undefined

So is division algorithm is still valid when divisor is zero? Please do tell any other restrictions(if any) for division algorithm.

Answer 1

You are dividing by $x-1$, which is not zero. The expression $p(x)=(x-1)q(x)+r(x)$ is correct for all values of $x$. If you choose a specific $p(x)$ you can see that by expanding the terms. There is no division in this expression. As it is true for all $x$ you can evaluate it at $x=1$ and it will still be true.

Answer 2

Polynomial division with the divisor being the zero polynomial does not fit Euclidean division of polynomials. See the Wikipedia article on Euclidean division of polynomials. The issue that I see is that the remainder must have a strictly lower degree than the divisor, whereas the degree of the zero polynomial is often defined as $-\infty$ for some good reasons. So a remainder cannot have even lower degree than that.