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: \[ -\frac{2y}{x + y} \]