1. **State the problem:** Divide the polynomial $p(x) = x^3 - x^2 + x - 1$ by $x - 1$ using synthetic division and find the quotient and remainder. 2. **Recall synthetic division:** When dividing by $x - c$, use $c$ as the divisor in synthetic division. 3. **Set up synthetic division:** Here, $c = 1$. Write the coefficients of $p(x)$: $1, -1, 1, -1$. 4. **Perform synthetic division:** - Bring down the first coefficient: $1$. - Multiply by $c$: $1 \times 1 = 1$. - Add to next coefficient: $-1 + 1 = 0$. - Multiply by $c$: $0 \times 1 = 0$. - Add to next coefficient: $1 + 0 = 1$. - Multiply by $c$: $1 \times 1 = 1$. - Add to last coefficient: $-1 + 1 = 0$. 5. **Write quotient and remainder:** The quotient coefficients are $1, 0, 1$, so quotient is $x^2 + 0x + 1 = x^2 + 1$. The remainder is $0$. 6. **Conclusion:** $$\frac{x^3 - x^2 + x - 1}{x - 1} = x^2 + 1$$ with remainder $0$. This means $x - 1$ divides $p(x)$ exactly.