zero rule of productZeroRuleOfProduct2005-03-06 04:32:112009-09-22 21:15:46Resultintegral domain% this is the default PlanetMath preamble. as your knowledge
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% define commands hereFor real and complex numbers, and more generally for elements of an integral domain, a product equals to zero if and only if at least one of the \PMlinkescapetext{factors} equals to zero.\, For two elements $a$ and $b$, we have
$$ab = 0 \quad \Leftrightarrow \quad a = 0 \, \lor \, b = 0. $$
For example, this rule can be used in solving polynomial equations:
$$x^3-x^2-2x+2 = 0$$
$$(x^3-x^2)+(-2x+2) = 0$$
$$x^2(x-1)-2(x-1) = 0$$
$$(x-1)(x^2-2) = 0$$
$$x-1 = 0 \,\lor\, x^2-2 = 0$$
$$x = 1 \,\lor\, x = \pm\sqrt{2}$$
The used sign ``$\lor$'' is the logical or.