Question

Express in simplest form: 2y2−2xyx2−y2
2
𝑦
2
−
2
𝑥
𝑦
𝑥
2
−
𝑦
2
(1 point)
Responses

2−2xyx
2
−
2
𝑥
𝑦
𝑥
the fraction with numerator 2 minus 2 x y and denominator x

−2xyx+y
−
2
𝑥
𝑦
𝑥
+
𝑦
the fraction with numerator negative 2 x y and denominator x plus y

−2x
−
2
𝑥
negative 2 over x

2xyx+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 \(x^2 - y^2\) is a difference of squares, which we can factor as:
\[
x^2 - y^2 = (x - y)(x + y)
\]

Now we have:
\[
\frac{2y(y - x)}{(x - y)(x + y)}
\]

Notice that \((y - x)\) is the negative of \((x - y)\):
\[
y - x = -(x - y)
\]

So we substitute this into our expression:
\[
\frac{2y(-(x - y))}{(x - y)(x + y)} = \frac{-2y(x - y)}{(x - y)(x + y)}
\]

Now, we can cancel out the \((x - y)\) term in the numerator and denominator (assuming \(x \neq y\)):
\[
\frac{-2y}{x + y}
\]

Thus, the simplified form of the expression is:
\[
\boxed{\frac{-2y}{x + y}}
\]