Question

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

Answer 1

`u^2-2u+1-v^2=(u-v-1)(u+v-1)`

Explanation:

The difference of squares identity can be written:

`a^2-b^2 = (a-b)(a+b)`

Use this with `a=(u-1)` and `b=v` as follows:

`u^2-2u+1-v^2=(u-1)^2-v^2`

`color(white)(u^2-2u+1-v^2)=((u-1)-v)((u-1)+v)`

`color(white)(u^2-2u+1-v^2)=(u-1-v)(u-1+v)`

`color(white)(u^2-2u+1-v^2)=(u-v-1)(u+v-1)`