Express in simplest form: 2y2−2xyx2−y2

Question

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