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

$$ab=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}⟺\mathit{\hspace{1em}}a=\mathrm{\hspace{0.17em}0}\vee b=\mathrm{\hspace{0.17em}0}.$$

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

$$x^3-x^2-2x+2=\mathrm{\hspace{0.33em}0}$$

$$(x^3-x^2)+(-2x+2)=\mathrm{\hspace{0.33em}0}$$

$$x^2(x-1)-2(x-1)=\mathrm{\hspace{0.33em}0}$$

$$(x-1)(x^2-2)=\mathrm{\hspace{0.33em}0}$$

$$x-1=\mathrm{\hspace{0.33em}0}\vee x^2-2=\mathrm{\hspace{0.33em}0}$$

$$x=\mathrm{\hspace{0.33em}1}\vee x=\pm \sqrt{2}$$

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