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[parent] zero rule of product (Result)

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

$\displaystyle ab = 0 \quad \Leftrightarrow \quad a = 0 \, \lor \, b = 0. $

For example, this rule can be used in solving polynomial equations:

$\displaystyle x^3-x^2-2x+2 = 0$
$\displaystyle (x^3-x^2)+(-2x+2) = 0$
$\displaystyle x^2(x-1)-2(x-1) = 0$
$\displaystyle (x-1)(x^2-2) = 0$
$\displaystyle x-1 = 0 \,\lor\, x^2-2 = 0$
$\displaystyle x = 1 \,\lor\, x = \pm\sqrt{2}$

The used sign “$ \lor$” is the logical or.



"zero rule of product" is owned by pahio.
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See Also: cancellation ring, Euler's derivation of the quartic formula, grouping method, grouping method for factoring polynomials

Other names:  product to zero rule

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Cross-references: logical or, equations, polynomial, product, integral domain, complex numbers, real
There are 2 references to this entry.

This is version 6 of zero rule of product, born on 2005-03-06, modified 2008-02-28.
Object id is 6848, canonical name is ZeroRuleOfProduct.
Accessed 4445 times total.

Classification:
AMS MSC: 13G05 (Commutative rings and algebras :: Integral domains)
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