i) \( \frac{2}{x^{2}(x-1)} \)

We wish to express   2/(x²(x – 1)) in partial fractions. Since the denominator factors as x²(x – 1), we assume a decomposition of the form   2/(x²(x – 1)) = A/x + B/x² + C/(x – 1). Step 1. Multiply both sides by x²(x – 1) to eliminate denominators:   2 = A·[x·(x – 1)] + B·(x – 1) + C·x². Step 2. Expand the right-hand side:   A·[x² – x] = A·x² – A·x,   B·(x – 1) = B·x – B,   and we already have C·x². Thus, combine like terms:   2 = (A + C)x² + (B – A)x – B. Step 3. Equate coefficients from both sides. On the left-hand side, there is no x² term or x term; it’s just the constant 2. Therefore, we have   Coefficient of x²: A + C = 0   (1)   Coefficient of x: B – A = 0    (2)   Constant term: –B = 2       (3) Step 4. Solve the system: From (3): –B = 2 ⟹ B = –2. From (2): B – A = 0 ⟹ –2 – A = 0 ⟹ A = –2. From (1): A + C = 0 ⟹ –2 + C = 0 ⟹ C = 2. Step 5. Write the final answer:   2/(x²(x – 1)) = (–2)/x + (–2)/x² + (2)/(x – 1) or more neatly,   2/(x²(x – 1)) = –2/x – 2/x² + 2/(x – 1). Thus, the partial fractions decomposition is complete.