How do you factor u^2-2u+1-v^2?

Sep 2, 2016

${u}^{2} - 2 u + 1 - {v}^{2} = \left(u - v - 1\right) \left(u + v - 1\right)$

Explanation:

The difference of squares identity can be written:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Use this with $a = \left(u - 1\right)$ and $b = v$ as follows:

${u}^{2} - 2 u + 1 - {v}^{2} = {\left(u - 1\right)}^{2} - {v}^{2}$

$\textcolor{w h i t e}{{u}^{2} - 2 u + 1 - {v}^{2}} = \left(\left(u - 1\right) - v\right) \left(\left(u - 1\right) + v\right)$

$\textcolor{w h i t e}{{u}^{2} - 2 u + 1 - {v}^{2}} = \left(u - 1 - v\right) \left(u - 1 + v\right)$

$\textcolor{w h i t e}{{u}^{2} - 2 u + 1 - {v}^{2}} = \left(u - v - 1\right) \left(u + v - 1\right)$