$$ 1 \leftarrow \quad $$ Use synthetic division to find the function values. Then check your work using a graphing calculator. $$ f(x)=3 x^{5}-3 x^{4}+3 x^{3}-x+8 \text {; find } f(18) \text { and } f(-3) \text {. } $$

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We are given 
$$
f(x)=3x^5-3x^4+3x^3-x+8
$$
with missing term for $$ x^2 $$ (so its coefficient is 0). The coefficients are written in order as:
$$
3,\quad -3,\quad 3,\quad 0,\quad -1,\quad 8.
$$

We will use synthetic division to evaluate $$ f(18) $$ and $$ f(-3) $$.

---

### Evaluating $$ f(18) $$

Set up synthetic division with $$ 18 $$:

$$
\begin{array}{r|rrrrrr}
18 & 3 & -3 & 3 & 0 & -1 & 8 \
 &   & 54 & 51 & 921 & 16578 & 298404 \
\hline
  & 3 & 51 & 921 & 16578 & 298403 & 5371262 \
\end{array}
$$

**Step-by-step:**

1. **Bring down** the first coefficient: $$ 3 $$.

2. **Multiply**: $$ 3 \times 18 = 54 $$.  
   **Add**: $$ -3 + 54 = 51 $$.

3. **Multiply**: $$ 51 \times 18 = 918 $$.  
   **Add**: $$ 3 + 918 = 921 $$.

4. **Multiply**: $$ 921 \times 18 = 16578 $$.  
   **Add**: $$ 0 + 16578 = 16578 $$.

5. **Multiply**: $$ 16578 \times 18 = 298404 $$.  
   **Add**: $$ -1 + 298404 = 298403 $$.

6. **Multiply**: $$ 298403 \times 18 = 5371254 $$.  
   **Add**: $$ 8 + 5371254 = 5371262 $$.

The last number is the remainder, so
$$
f(18) = 5371262.
$$

---

### Evaluating $$ f(-3) $$

Now set up synthetic division with $$ -3 $$:

$$
\begin{array}{r|rrrrrr}
-3 & 3 & -3 & 3 & 0 & -1 & 8 \
 &   & -9 & 36 & -117 & 351 & -1050 \
\hline
  & 3 & -12 & 39 & -117 & 350 & -1042 \
\end{array}
$$

**Step-by-step:**

1. **Bring down** the first coefficient: $$ 3 $$.

2. **Multiply**: $$ 3 \times -3 = -9 $$.  
   **Add**: $$ -3 + (-9) = -12 $$.

3. **Multiply**: $$ -12 \times -3 = 36 $$.  
   **Add**: $$ 3 + 36 = 39 $$.

4. **Multiply**: $$ 39 \times -3 = -117 $$.  
   **Add**: $$ 0 + (-117) = -117 $$.

5. **Multiply**: $$ -117 \times -3 = 351 $$.  
   **Add**: $$ -1 + 351 = 350 $$.

6. **Multiply**: $$ 350 \times -3 = -1050 $$.  
   **Add**: $$ 8 + (-1050) = -1042 $$.

The last number is the remainder, so
$$
f(-3) = -1042.
$$

---

Thus, the computed values are:
$$
f(18) = 5371262 \quad \text{and} \quad f(-3) = -1042.
$$
$$ f(18) = 5,371,262 $$ and $$ f(-3) = -1,042 $$.

Use synthetic division to find the function values. Then check your work using a graphing calculator.

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