1. **State the problem:** We need to divide the polynomial $x^3 - 2x^2 + 1$ by the binomial $x - 1$. 2. **Recall the formula:** Polynomial division can be done using long division or synthetic division. Here, we use long division. 3. **Set up the division:** Divide $x^3 - 2x^2 + 0x + 1$ by $x - 1$. 4. **Divide the leading terms:** $\frac{x^3}{x} = x^2$. 5. **Multiply and subtract:** Multiply $x^2(x - 1) = x^3 - x^2$. Subtract from the original polynomial: $$\begin{aligned} (x^3 - 2x^2 + 0x + 1) - (x^3 - x^2) &= x^3 - 2x^2 + 0x + 1 - x^3 + x^2 \\ &= -x^2 + 0x + 1 \end{aligned}$$ 6. **Repeat division:** Divide leading terms $\frac{-x^2}{x} = -x$. 7. **Multiply and subtract:** Multiply $-x(x - 1) = -x^2 + x$. Subtract: $$\begin{aligned} (-x^2 + 0x + 1) - (-x^2 + x) &= -x^2 + 0x + 1 + x^2 - x \\ &= -x + 1 \end{aligned}$$ 8. **Repeat division:** Divide leading terms $\frac{-x}{x} = -1$. 9. **Multiply and subtract:** Multiply $-1(x - 1) = -x + 1$. Subtract: $$\begin{aligned} (-x + 1) - (-x + 1) &= -x + 1 + x - 1 = 0 \end{aligned}$$ 10. **Conclusion:** The quotient is $x^2 - x - 1$ and the remainder is $0$. **Final answer:** $$\frac{x^3 - 2x^2 + 1}{x - 1} = x^2 - x - 1$$