Simplify the algebraic expression. \( \frac{x^{2}-2 x y+y^{2}}{\frac{x}{y}-\frac{y}{x}} \) \( \frac{x^{2}-2 x y+y^{2}}{x}=\square \)

Simplify the expression by following steps: - step0: : \(x^{2}-2xy+y^{2}\) Expand the expression \( \frac{x}{y}-\frac{y}{x} \) Simplify the expression by following steps: - step0: Subtract the terms: \(\frac{x}{y}-\frac{y}{x}\) - step1: Reduce fractions to a common denominator: \(\frac{x\times x}{yx}-\frac{y\times y}{xy}\) - step2: Rewrite the expression: \(\frac{x\times x}{yx}-\frac{y\times y}{yx}\) - step3: Transform the expression: \(\frac{x\times x-y\times y}{yx}\) - step4: Multiply the terms: \(\frac{x^{2}-y\times y}{yx}\) - step5: Multiply the terms: \(\frac{x^{2}-y^{2}}{yx}\) Expand the expression \( \frac{x^{2}-2xy+y^{2}}{\frac{x^{2}-y^{2}}{yx}} \) Simplify the expression by following steps: - step0: Multiply the terms: \(\frac{x^{2}-2xy+y^{2}}{\frac{x^{2}-y^{2}}{yx}}\) - step1: Multiply by the reciprocal: \(\left(x^{2}-2xy+y^{2}\right)\times \frac{yx}{x^{2}-y^{2}}\) - step2: Rewrite the expression: \(\left(x-y\right)^{2}\times \frac{yx}{\left(x-y\right)\left(x+y\right)}\) - step3: Reduce the fraction: \(\left(x-y\right)\times \frac{yx}{x+y}\) - step4: Multiply the terms: \(\frac{\left(x-y\right)yx}{x+y}\) - step5: Multiply the terms: \(\frac{yx\left(x-y\right)}{x+y}\) - step6: Multiply the terms: \(\frac{yx^{2}-y^{2}x}{x+y}\) To simplify the algebraic expression \[ \frac{x^{2}-2xy+y^{2}}{\frac{x}{y}-\frac{y}{x}}, \] we will follow these steps: 1. **Simplify the numerator**: The numerator is \(x^{2} - 2xy + y^{2}\). This expression can be recognized as a perfect square trinomial, which factors to: \[ (x - y)^{2}. \] 2. **Simplify the denominator**: The denominator is \(\frac{x}{y} - \frac{y}{x}\). To combine these fractions, we find a common denominator: \[ \frac{x^{2} - y^{2}}{yx}. \] 3. **Combine the simplified numerator and denominator**: Now we can rewrite the original expression as: \[ \frac{(x - y)^{2}}{\frac{x^{2} - y^{2}}{yx}}. \] 4. **Simplify the overall expression**: Dividing by a fraction is equivalent to multiplying by its reciprocal: \[ \frac{(x - y)^{2} \cdot yx}{x^{2} - y^{2}}. \] 5. **Recognize that \(x^{2} - y^{2}\) can be factored**: The expression \(x^{2} - y^{2}\) can be factored as \((x - y)(x + y)\). Thus, we have: \[ \frac{(x - y)^{2} \cdot yx}{(x - y)(x + y)}. \] 6. **Cancel common factors**: We can cancel one \((x - y)\) from the numerator and denominator: \[ \frac{(x - y) \cdot yx}{x + y}. \] Thus, the simplified expression is: \[ \frac{(x - y)yx}{x + y}. \] So, the final answer is: \[ \frac{(x - y)yx}{x + y} = \square. \]