Question

factor the expression u*(u-v)+v*(v-u)

Answer 1

To factor the expression \( u(u - v) + v(v - u) \), let's start by rewriting it:

\[
u(u - v) + v(v - u) = u^2 - uv + v^2 - vu
\]

Notice that \( -uv \) and \( -vu \) are the same term. Therefore, we can simplify further:

\[
u^2 - uv + v^2 - uv = u^2 - 2uv + v^2
\]

Now we recognize that \( u^2 - 2uv + v^2 \) can be factored as a perfect square:

\[
u^2 - 2uv + v^2 = (u - v)^2
\]

So, the original expression \( u(u - v) + v(v - u) \) factors to:

\[
(u - v)^2
\]

Thus, the final factored form is:

\[
\boxed{(u - v)^2}
\]