$P(x)$ with arbitrary degree and coefficients divided $(x-r_1)$ by synthetic division, should the remainder be equal to $0$?

Question

Generally: Suppose we are given a polynomial of arbitrary degree $P(x)$ with arbitrary coefficients $a_1,a_2,\cdots,a_n$, and we divide $P(x)$ by $(x-r_1)$ where $r_1$ is one of its roots, should we expect a remainder when performing synthetic division?

If there is a remainder, say $3a_2+a_1+3$, should this be set to equal $0$?

Answer 1

An easy claim you can prove is:

Claim: When we divide any polynomial $\;P(x)\;$ by a linear polynomial of the form $\;x-r\;$ , the remainder is $\;P(r)\;$ .

And thus if $\;r\;$ is a root of $\;P(x)\;$ the remainder of the wanted division is $\;P(r)=0\;$ .