1. **State the problem:** We want to express the rational function $$\frac{x - 2}{x^2 - 2x + 1}$$ in partial fractions. 2. **Factor the denominator:** Note that $$x^2 - 2x + 1 = (x - 1)^2$$. 3. **Set up the partial fraction decomposition:** Since the denominator is a repeated linear factor, the decomposition takes the form: $$\frac{x - 2}{(x - 1)^2} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2}$$ 4. **Multiply both sides by the denominator to clear fractions:** $$x - 2 = A(x - 1) + B$$ 5. **Expand and simplify:** $$x - 2 = A x - A + B$$ 6. **Group like terms:** $$x - 2 = A x + (B - A)$$ 7. **Equate coefficients of like terms:** - Coefficient of $$x$$: $$1 = A$$ - Constant term: $$-2 = B - A$$ 8. **Solve for constants:** From $$1 = A$$, we get $$A = 1$$. Substitute into the constant term equation: $$-2 = B - 1 \implies B = -1$$ 9. **Write the final partial fraction decomposition:** $$\frac{x - 2}{(x - 1)^2} = \frac{1}{x - 1} - \frac{1}{(x - 1)^2}$$ This expresses the original function as a sum of simpler rational functions with denominators of lower degree, which is the goal of partial fraction decomposition.