Express in simplest form: 2y2−2xyx2−y2 .(1 point) Responses 2yx+y Start Fraction 2 y over x plus y End Fraction −2x Start Fraction negative 2 over x End Fraction −2yx+y Start Fraction negative 2 y over x plus y End Fraction 2−2yx

Answer 1

To simplify the expression \( \frac{2y^2 - 2xy}{x^2 - y^2} \), we will first factor both the numerator and the denominator.

1. Factor the numerator:

   The numerator can be factored as: \[ 2y^2 - 2xy = 2(y^2 - xy) = 2y(y - x) \]

2. Factor the denominator:

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

3. Put it all together:

   After factoring, the expression becomes: \[ \frac{2y(y - x)}{(x - y)(x + y)} \]

   Note that \( (x - y) = -1(y - x) \), so we can rewrite the expression: \[ \frac{2y(y - x)}{-1(y - x)(x + y)} = \frac{-2y}{(x + y)} \]

4. Final simplified form:

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

Therefore, the simplest form of the expression \( \frac{2y^2 - 2xy}{x^2 - y^2} \) is: \[ \frac{-2y}{x + y} \]