Answer
Remainder Theorem - If the polynomial $P(x)$ is divided by $x-c$, then the remainder is the value $P(c)$
Factor Theorem - $c$ is a zero of $P$ if and only if $x-c$ is a factor of $P(x)$
Rational Zeroes Theorem - If the polynomial $P(x)=a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0$ has integer coeficients (where $a_n\ne0$ and $a_0\ne0$), then every rational zero of $P$ is of the form :
$$\frac{p}{q}$$ where $p$ and $q$ are integers and
$p$ is a factor of the constant coefficient $a_0$
$q$ is a factor of the leading coefficient $a_n$
Work Step by Step
As also explained in the chapter earlier, the theorems are based on the following ideas :
Remainder Theorem - If the polynomial $P(x)$ is divided by $x-c$, then the remainder is the value $P(c)$
Factor Theorem - $c$ is a zero of $P$ if and only if $x-c$ is a factor of $P(x)$
Rational Zeroes Theorem - If the polynomial $P(x)=a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0$ has integer coeficients (where $a_n\ne0$ and $a_0\ne0$), then every rational zero of $P$ is of the form :
$$\frac{p}{q}$$ where $p$ and $q$ are integers and
$p$ is a factor of the constant coefficient $a_0$
$q$ is a factor of the leading coefficient $a_n$