# zero rule of product

For 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 equals to zero.  For two elements $a$ and $b$, we have

 $ab\;=\;0\quad\Longleftrightarrow\quad a\,=\,0\;\lor\;b\,=\,0.$

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.

Title zero rule of product ZeroRuleOfProduct 2013-03-22 15:06:46 2013-03-22 15:06:46 pahio (2872) pahio (2872) 11 pahio (2872) Result msc 13G05 product to zero rule CancellationRing EulersDerivationOfTheQuarticFormula GroupingMethodForFactorizingPolynomials HyperbolasOrthogonalToEllipses