Question

Express in simplest form:  2y2−2xyx2−y2 .(1 point)Responses2−2yxStart Fraction 2 minus 2 y over x End Fraction−2yx+yStart Fraction negative 2 y over x plus y End Fraction−2xStart Fraction negative 2 over x End Fraction2yx+y

Answer 1

To simplify the expression \(\frac{2y^2 - 2xy}{x^2 - y^2}\), we can start by factoring both the numerator and the denominator.

1. **Factor the numerator**:
\[
2y^2 - 2xy = 2(y^2 - xy) = 2y(y - x)
\]

2. **Factor the denominator**:
The denominator \(x^2 - y^2\) can be factored using the difference of squares formula:
\[
x^2 - y^2 = (x - y)(x + y)
\]

Now, we can rewrite the expression with the factored forms:
\[
\frac{2y(y - x)}{(x - y)(x + y)}
\]

Notice that \(y - x = -(x - y)\), so we can substitute:
\[
\frac{2y(y - x)}{(x - y)(x + y)} = \frac{2y(-1)(x - y)}{(x - y)(x + y)}
\]

The \((x - y)\) terms in the numerator and denominator will cancel out (as long as \(x \neq y\)):
\[
= \frac{-2y}{x + y}
\]

Thus, the expression simplifies to:
\[
-\frac{2y}{x + y}
\]

The final answer in simplest form is:
\[
-\frac{2y}{x + y}
\]