$\frac{x}{x - 1} + \frac{1}{x + 1} + \frac{2x}{1 - x^2} =$ ?

### Concept & Formula This problem requires simplifying rational expressions by finding a common denominator. The key is recognizing the **Difference of Squares** identity in the third term's denominator. $$A^2 - B^2 = (A - B)(A + B)$$ ### Step-by-Step Solution Given expression: $\frac{x}{x - 1} + \frac{1}{x + 1} + \frac{2x}{1 - x^2}$ **Step 1: Standardize the denominators** Notice the denominator of the third term is $1 - x^2$. We can rewrite this to match the other denominators by factoring out a negative sign: $1 - x^2 = -(x^2 - 1) = -(x - 1)(x + 1)$ Rewrite the original expression with this substitution: $\frac{x}{x - 1} + \frac{1}{x + 1} - \frac{2x}{(x - 1)(x + 1)}$ **Step 2: Find a common denominator** The least common multiple for the denominators is $(x - 1)(x + 1)$. Adjust the first two fractions so they all share this common denominator: - Multiply the first term by $\frac{x + 1}{x + 1}$ - Multiply the second term by $\frac{x - 1}{x - 1}$ $= \frac{x(x + 1)}{(x - 1)(x + 1)} + \frac{1(x - 1)}{(x - 1)(x + 1)} - \frac{2x}{(x - 1)(x + 1)}$ **Step 3: Combine and simplify** Combine all numerators over the common denominator: $= \frac{x^2 + x + x - 1 - 2x}{(x - 1)(x + 1)}$ Group the like terms in the numerator: $= \frac{x^2 + 2x - 1 - 2x}{(x - 1)(x + 1)}$ $= \frac{x^2 - 1}{(x - 1)(x + 1)}$ Since $x^2 - 1$ factors to $(x - 1)(x + 1)$: $= \frac{x^2 - 1}{x^2 - 1} = 1$ ### Exam Strategy & Shortcut **Value Assumption:** To solve this rapidly during an exam, assign a small integer value to $x$. Let's pick $x = 2$ (avoid $x = 1$ or $x = -1$ to prevent dividing by zero). Substitute $x = 2$ into the expression: $= \frac{2}{2 - 1} + \frac{1}{2 + 1} + \frac{2(2)}{1 - 2^2}$ $= \frac{2}{1} + \frac{1}{3} + \frac{4}{1 - 4}$ $= 2 + \frac{1}{3} - \frac{4}{3}$ $= 2 - \frac{3}{3}$ $= 2 - 1 = 1$. ### Common Pitfall Failing to flip the sign on the third fraction when converting $1 - x^2$ to $x^2 - 1$. If you forget to change the $+$ to a $-$, the numerator becomes $x^2 + 4x - 1$, which does not simplify, causing students to get stuck and waste time. ### Final Answer Therefore, the correct answer is **1**.