Question

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