Use the remainder theorem to evaluate the polynomial for the given values of $$ x $$. $$ f(x)=3 x^{3}-x^{2}-18 x+6 $$

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**Step 1. Statement of the Remainder Theorem**

The Remainder Theorem tells us that if a polynomial $$ f(x) $$ is divided by $$(x - a)$$, the remainder is equal to $$ f(a) $$. In other words,

$$
f(a) = \text{remainder when } f(x) \text{ is divided by } (x-a).
$$

**Step 2. Evaluating $$ f(x)=3x^3-x^2-18x+6 $$ at a Specific Value**

Since the problem asks us to “evaluate the polynomial for the given values of $$ x $$” using the Remainder Theorem, we assume that for any number $$ a $$ provided, the remainder when dividing by $$(x-a)$$ is exactly $$ f(a) $$.

Below we show how to evaluate $$ f(x) $$ for two sample values, $$ x=1 $$ and $$ x=-2 $$.

---

**Example 1. Evaluating $$ f(1) $$**

We need to find the remainder when dividing $$ f(x)=3x^3-x^2-18x+6 $$ by $$(x-1)$$.

*Using synthetic division:*

1. Write the coefficients: $$ 3, -1, -18, 6 $$.
2. Place the value $$ 1 $$ (from $$ x-1 $$) to the left.

$$
\begin{array}{r|rrrr}
1 & 3 & -1 & -18 & 6 \
  &   & 3  & 2   & -16 \ \hline
  & 3 & 2  & -16 & -10 \
\end{array}
$$

- **Bring down** the first coefficient $$ 3 $$.
- Multiply $$ 3 $$ by $$ 1 $$ to get $$ 3 $$; add to $$-1$$ to get $$ 2 $$.
- Multiply $$ 2 $$ by $$ 1 $$ to get $$ 2 $$; add to $$-18$$ to get $$-16$$.
- Multiply $$-16$$ by $$ 1 $$ to get $$-16$$; add to $$ 6 $$ to get $$-10$$.

The final number, $$-10$$, is the remainder; hence,

$$
f(1) = -10.
$$

---

**Example 2. Evaluating $$ f(-2) $$**

Now, we evaluate $$ f(x) $$ for $$ x=-2 $$. This is equivalent to finding the remainder when dividing $$ f(x) $$ by $$(x - (-2)) = (x+2)$$.

*Using synthetic division:*

1. Write the coefficients: $$ 3, -1, -18, 6 $$.
2. Place the value $$-2$$ to the left.

$$
\begin{array}{r|rrrr}
-2 & 3 & -1 & -18 & 6 \
   &   & -6 & 14  & 8 \ \hline
   & 3 & -7 & -4  & 14 \
\end{array}
$$

- **Bring down** the first coefficient $$ 3 $$.
- Multiply $$ 3 $$ by $$-2$$ to get $$-6$$; add to $$-1$$ to get $$-7$$.
- Multiply $$-7$$ by $$-2$$ to get $$ 14 $$; add to $$-18$$ to get $$-4$$.
- Multiply $$-4$$ by $$-2$$ to get $$ 8 $$; add to $$ 6 $$ to get $$14$$.

Thus, the remainder is $$ 14 $$, meaning that

$$
f(-2) = 14.
$$

---

**Conclusion**

Using the Remainder Theorem, we have:
$$
\begin{aligned}
f(1) &= -10,\[1mm]
f(-2) &= 14.
\end{aligned}
$$

For any other value, say $$ x = a $$, you can perform synthetic division (or direct substitution) to find that the remainder when dividing $$ f(x) $$ by $$(x-a)$$ is exactly $$ f(a) $$.
To evaluate the polynomial $$ f(x) = 3x^3 - x^2 - 18x + 6 $$ at specific values using the Remainder Theorem: 1. **For $$ x = 1 $$:** - Remainder = $$ f(1) = -10 $$ 2. **For $$ x = -2 $$:** - Remainder = $$ f(-2) = 14 $$ For other values of $$ x $$, perform synthetic division or direct substitution to find $$ f(a) $$.

Use the remainder theorem to evaluate the polynomial for the given values of \( x \). \[ f(x)=3 x^{3}-x^{2}-18 x+6 \]

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