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# 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 factors 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.

Related:

CancellationRing, EulersDerivationOfTheQuarticFormula, GroupingMethodForFactorizingPolynomials, HyperbolasOrthogonalToEllipses

Synonym:

product to zero rule

Major Section:

Reference

Type of Math Object:

Result

Parent:

## Mathematics Subject Classification

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