Express in simplest form:

Question

2
y
2
−
2
x
y
x
2
−
y
2
(1 point)
Responses

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

2
x
y
x
+
y
the fraction with numerator 2 x y and denominator x plus y

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

−
2
x

Answer 1

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

1. Factor the numerator: The numerator \(2y^2 - 2xy\) can be factored as follows: \[ 2y^2 - 2xy = 2y(y - x) \]

2. Factor the denominator: The denominator \(x^2 - y^2\) is a difference of squares and can be factored as: \[ x^2 - y^2 = (x - y)(x + y) \]

Thus, we have: \[ \frac{2y^2 - 2xy}{x^2 - y^2} = \frac{2y(y - x)}{(x - y)(x + y)} \]

Recognizing that \(y - x = -(x - y)\), we can rewrite the expression: \[ = \frac{2y(y - x)}{-(x - y)(x + y)} = \frac{-2y(y - x)}{(x - y)(x + y)} \]

3. Simplify: Now we can simplify further: \[ = \frac{-2y}{x + y} \]

So, the simplest form of the expression is: \[ \frac{-2y}{x + y} \]

Therefore, the response is: −2y/(x + y).