To evaluate the function $f(x) = x^3 + x^2 - 3x + 9$ at $x = -4$ using synthetic division, we will follow these steps:

Set Up the Synthetic Division:

1. Write the coefficients of the polynomial $f(x) = x^3 + x^2 - 3x + 9$. The coefficients are $1, 1, -3, 9$.

2. Set up the synthetic division by writing $-4$ (the value of $x$) to the left of a vertical bar and the coefficients $1, 1, -3, 9$ to the right.

Perform the Synthetic Division:

We perform synthetic division by carrying out the following calculations:

-4 & 1 & 1 & -3 & 9 \\ & & -4 & 12 & -36 \\ \hline & 1 & -3 & 9 & -27 \\ \end{array}$$ Let's go through this step-by-step: 1. **Bring down** the first coefficient, $$1$$. 2. **Multiply** this coefficient by $$-4$$ (the value of $$x$$) to get $$-4$$, and **write** it under the next coefficient $$1$$. 3. **Add** the column: $$1 + (-4) = -3$$. Write $$-3$$ below. 4. **Multiply** $$-3$$ by $$-4$$ to get $$12$$, and **write** it under the next coefficient $$-3$$. 5. **Add** the column: $$-3 + 12 = 9$$. Write $$9$$ below. 6. **Multiply** $$9$$ by $$-4$$ to get $$-36$$, and **write** it under the next coefficient $$9$$. 7. **Add** the column: $$9 + (-36) = -27$$. Write $$-27$$ below. The last number, $$-27$$, is the remainder, which is also the value of $$f(-4)$$. ### Conclusion: The value of the function $$f(x) = x^3 + x^2 - 3x + 9$$ at $$x = -4$$ is $$f(-4) = -27$$. Would you like more details on synthetic division, or do you have any questions? Here are five related questions: 1. How do you interpret the remainder in synthetic division when evaluating a polynomial? 2. Can synthetic division be used to find roots of a polynomial? How? 3. What happens when the remainder in synthetic division is zero? 4. How would synthetic division change if evaluating at $$x = 4$$ instead of $$x = -4$$? 5. How can synthetic division be used to factor polynomials? **Tip:** Remember that synthetic division is particularly useful for quickly evaluating polynomials at specific values and finding roots when $$x - c$$ is a factor of the polynomial.