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Version 5 |
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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 \PMlinkescapetext{factors} equals to zero.\, For two elements $a$ and $b$, we have
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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 $b$ of the integral domain, we have
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| $$ab = 0 \quad \Leftrightarrow \quad a = 0 \, \lor \, b = 0. $$ |
$$ab = 0 \quad \Leftrightarrow \quad a = 0 \, \lor \, b = 0. $$ |
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| For example, this rule can be used in solving polynomial equations: |
For example, this rule can be used in solving polynomial equations: |
| $$x^3-x^2-2x+2 = 0$$ |
$$x^3-x^2-2x+2 = 0$$ |
| $$(x^3-x^2)+(-2x+2) = 0$$ |
$$(x^3-x^2)+(-2x+2) = 0$$ |
| $$x^2(x-1)-2(x-1) = 0$$ |
$$x^2(x-1)-2(x-1) = 0$$ |
| $$(x-1)(x^2-2) = 0$$ |
$$(x-1)(x^2-2) = 0$$ |
| $$x-1 = 0 \,\lor\, x^2-2 = 0$$ |
$$x-1 = 0 \,\lor\, x^2-2 = 0$$ |
| $$x = 1 \,\lor\, x = \pm\sqrt{2}$$ |
$$x = 1 \,\lor\, x = \pm\sqrt{2}$$ |
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| The used sign ``$\lor$'' is the logical or. |
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