Is it possible to prove that a polynomial with a root at $x_0$ can be written as $p(x)=(x-x_0)q(x)$ with Taylor Polynomials?

Question

I know that if $p$ is a polynomial of degree at most $n$ and $x_0$ is any point, then the $n$th Taylor Polynomial for $p$ at $x_0$ is $p$ itself. My textbook immediately says after this that you can use this to prove the assertion. How? I do not see the connection.

Answer 1

Hint:

Write the $n^{th}$ Taylor polynomial for $p$ about the point $x_0$, and evaluate the taylor polynomial at $x_0$. Using the fact that $p(x_0) = 0$, and the fact that the Taylor polynomial equals $p$, what can you say about the constant term of the Taylor polynomial? Can you see how to conclude from here?