Question: You want to divide $x^3 - x^2 + 12$ by $x - 3$ by using synthetic division.
Which of the following is the correct setup?
-3 | 1 -1 0 12
3 | 1 -1 12
-3 | 1 -1 12
3 | 1 -1 0 12
1. **State the problem:** We want to divide the polynomial $$x^3 - x^2 + 0x + 12$$ by the binomial $$x - 3$$ using synthetic division.
2. **Identify the divisor root:** For synthetic division, we use the root of the divisor $$x - 3 = 0$$ which is $$x = 3$$.
3. **Set up synthetic division:** Write the coefficients of the dividend polynomial in order: $$1$$ (for $$x^3$$), $$-1$$ (for $$x^2$$), $$0$$ (for $$x$$), and $$12$$ (constant term).
4. **Correct synthetic division setup:** The setup should have the root $$3$$ on the left and the coefficients $$1, -1, 0, 12$$ on the right:
$$
3 \mid 1 \quad -1 \quad 0 \quad 12
$$
5. **Check the options:**
- Option 1: $$-3 \mid 1 \quad -1 \quad 0 \quad 12$$ (incorrect root)
- Option 2: $$3 \mid 1 \quad -1 \quad 12$$ (missing the zero coefficient)
- Option 3: $$-3 \mid 1 \quad -1 \quad 12$$ (incorrect root and missing zero)
- Option 4: $$3 \mid 1 \quad -1 \quad 0 \quad 12$$ (correct)
**Final answer:** The correct setup is $$3 \mid 1 \quad -1 \quad 0 \quad 12$$.