Chromatic polynomial product over components ($p_G(x)={\prod }_j{p}_{G_j}(x)$)

For connected components of a graph $G_1,\dots ,G_t$ of a finite graph $G$, choices of proper vertex colourings on different connected components of a graph are independent. The multiplication principle gives $p_G(x)={\prod }_j{p}_{G_j}(x)$. Each nonempty factor is divisible by $x$, hence $x^t$ divides the chromatic polynomial. The vanishing constant term follows by mathematical induction from the deletion-contraction recurrence for the chromatic polynomial, starting with $p_H(x)=x^{|V(H)|}$ for a nonempty graph without edges.