Question
Express in simplest form: 2y2−2xyx2−y2
(1 point)
Responses
2−2yx
the fraction with numerator 2 minus 2 y and denominator x
−2x
negative 2 over x
−2yx+y
the fraction with numerator negative 2 y and denominator x plus y
2yx+y
(1 point)
Responses
2−2yx
the fraction with numerator 2 minus 2 y and denominator x
−2x
negative 2 over x
−2yx+y
the fraction with numerator negative 2 y and denominator x plus y
2yx+y
Answers
GPT-4o mini
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 can be factored as:
\[
x^2 - y^2 = (x - y)(x + y)
\]
Putting these factored forms into the expression gives:
\[
\frac{2y(y - x)}{(x - y)(x + y)}
\]
We can notice that \(y - x\) is equivalent to \(-(x - y)\), so we can rewrite the numerator:
\[
(y - x) = -(x - y)
\]
Thus, substituting that back into our expression gives:
\[
\frac{2y(-1)(x - y)}{(x - y)(x + y)} = \frac{-2y(x - y)}{(x - y)(x + y)}
\]
We can cancel out \(x - y\) from the numerator and denominator, assuming \(x \neq y\):
\[
= \frac{-2y}{x + y}
\]
This is the simplest form of the original expression. Therefore, the answer is:
\[
\frac{-2y}{x + y}
\]
None of the provided responses match the simplified expression directly. However, to check if we have misinterpreted the problem or need to choose from the options, let's ensure our interpretation of the question met the correct calculations and simplifications.
If the responses provided were seeking alternative interpretations or formats, please verify. Otherwise, we arrive at the simplest form being \(\frac{-2y}{x + y}\).
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 can be factored as:
\[
x^2 - y^2 = (x - y)(x + y)
\]
Putting these factored forms into the expression gives:
\[
\frac{2y(y - x)}{(x - y)(x + y)}
\]
We can notice that \(y - x\) is equivalent to \(-(x - y)\), so we can rewrite the numerator:
\[
(y - x) = -(x - y)
\]
Thus, substituting that back into our expression gives:
\[
\frac{2y(-1)(x - y)}{(x - y)(x + y)} = \frac{-2y(x - y)}{(x - y)(x + y)}
\]
We can cancel out \(x - y\) from the numerator and denominator, assuming \(x \neq y\):
\[
= \frac{-2y}{x + y}
\]
This is the simplest form of the original expression. Therefore, the answer is:
\[
\frac{-2y}{x + y}
\]
None of the provided responses match the simplified expression directly. However, to check if we have misinterpreted the problem or need to choose from the options, let's ensure our interpretation of the question met the correct calculations and simplifications.
If the responses provided were seeking alternative interpretations or formats, please verify. Otherwise, we arrive at the simplest form being \(\frac{-2y}{x + y}\).