= Chromatic polynomial product over components
{title2=$p_G(x)=\prod_jp_{G_j}(x)$}
For <connected components of a graph> $G_1,\ldots,G_t$ of a finite <graph> $G$, choices of <proper vertex colourings> on different <connected components of a graph> are independent. The <multiplication principle> gives $p_G(x)=\prod_jp_{G_j}(x)$. 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>.
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