My book said it is easy to see this: "If a polynomial $P$ is the product of two factors $P =P_1 .P_2$, then, $C_P$, the curve defined by $P$, is equal to $C_{P_1} \cup C_{P_2}$, where $C_{P_i}$ is the curve defined by $P_i, i =1,2$".
But I'm not sure of this. For example, $x^3 = x^2 . x$, but the graph of $x^3$ isn't the union of a line and a parabola...
You're looking at the zero set of $0=P$. For example if $P=x^2-y^2 = (x+y)(x-y)$, then the curve defined by $P$ is the union of the two lines $0=x+y$ and $0=x-y$.