I am reading a definition in my Pre-Calculus book but I am a little but confused, the definition states:
Suppose $p$ is a nonzero polynomial with at least one (real) zero. Then
*There exist real numbers $r_1$,$r_2$,...,$r_m$ and a polynomial G such that G has no (real) zeroes and $p(x)=(x-r_1)(x-r_2)...(x-r_m)G(x)$ for every real number $x$;
*each of the numbers $r_1$,$r_2$,...,$r_m$ is a zero of $p$;
*$p$ has no zeros other than $r_1$,$r_2$,...,$r_m$.
I understand why $p(x)=(x-r_1)(x-r_2)...(x-r_m)$ makes sense but I am having trouble understanding why there is a polynomial $G(x)$ that has no (real) zeroes at the end. Could someone please explain this to me? I am really confused.