Synthetic Division Setup A308B1

1. **State the problem:** We want to divide the polynomial $$x^3 + 4x^2 + x - 6$$ by $$x - 2$$ using synthetic division. 2. **Recall the synthetic division setup:** - The divisor is of the form $$x - c$$, so the number used in synthetic division is $$c$$. - Here, the divisor is $$x - 2$$, so $$c = 2$$. 3. **Identify the coefficients of the dividend polynomial:** - The polynomial $$x^3 + 4x^2 + x - 6$$ has coefficients $$1, 4, 1, -6$$. 4. **Set up synthetic division:** - Use $$c = 2$$ on the left. - Use the coefficients $$1, 4, 1, -6$$ on the right. 5. **Check the options:** - a) uses $$-2$$ and coefficients $$1,4,1,-6$$ (incorrect $$c$$). - b) uses $$-2$$ and coefficients $$3,2,1,-6$$ (incorrect $$c$$ and coefficients). - c) uses $$2$$ and coefficients $$3,2,1,-6$$ (correct $$c$$ but wrong coefficients). - d) uses $$2$$ and coefficients $$1,4,1,-6$$ (correct $$c$$ and correct coefficients). 6. **Conclusion:** The correct synthetic division setup is option d.) $$2 \mid 1\ 4\ 1\ -6$$. **Note:** The user states the correct answer is c.) with $$2 \mid 3\ 2\ 1\ -6$$, but this is inconsistent with the polynomial given. The coefficients of $$x^3 + 4x^2 + x - 6$$ are $$1,4,1,-6$$, not $$3,2,1,-6$$. **Final answer:** d.) $$2 \mid 1\ 4\ 1\ -6$$ is the correct setup for synthetic division of $$x^3 + 4x^2 + x - 6$$ by $$x - 2$$.