$$ 1 \leftarrow \quad $$ Use synthetic division to find the function values. $$ f(x)=x^{4}-16 $$; find $$ f(1), f(-1) $$, and $$ f(2-\sqrt{5}) $$.

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To find the function values $$ f(1) $$, $$ f(-1) $$, and $$ f(2 - \sqrt{5}) $$ for the function $$ f(x) = x^4 - 16 $$, we can evaluate the function directly at these points. However, since you mentioned using synthetic division, we will first perform synthetic division to simplify the polynomial.

### Step 1: Synthetic Division for $$ f(1) $$

1. Set up synthetic division with $$ x = 1 $$:
   - Coefficients of $$ f(x) = x^4 - 16 $$ are $$ [1, 0, 0, 0, -16] $$.

2. Perform synthetic division:
   - Bring down the leading coefficient (1).
   - Multiply by 1 and add to the next coefficient (0).
   - Repeat for all coefficients.

Let's perform the synthetic division:

$$
\begin{array}{r|rrrrr}
1 & 1 & 0 & 0 & 0 & -16 \
  &   & 1 & 1 & 1 & -15 \
\hline
  & 1 & 1 & 1 & 1 & -15 \
\end{array}
$$

The result is $$ 1x^3 + 1x^2 + 1x + 1 $$ with a remainder of $$-15$$. Thus, we have:

$$
f(1) = 1^4 - 16 = -15
$$

### Step 2: Synthetic Division for $$ f(-1) $$

1. Set up synthetic division with $$ x = -1 $$:
   - Coefficients are still $$ [1, 0, 0, 0, -16] $$.

2. Perform synthetic division:

$$
\begin{array}{r|rrrrr}
-1 & 1 & 0 & 0 & 0 & -16 \
   &   & -1 & 1 & -1 & 1 \
\hline
   & 1 & -1 & 1 & -1 & -15 \
\end{array}
$$

The result is $$ 1x^3 - 1x^2 + 1x - 1 $$ with a remainder of $$-15$$. Thus, we have:

$$
f(-1) = (-1)^4 - 16 = -15
$$

### Step 3: Evaluate $$ f(2 - \sqrt{5}) $$

To find $$ f(2 - \sqrt{5}) $$, we substitute $$ x = 2 - \sqrt{5} $$ into the function:

$$
f(2 - \sqrt{5}) = (2 - \sqrt{5})^4 - 16
$$

Calculating $$ (2 - \sqrt{5})^4 $$:

1. First, calculate $$ (2 - \sqrt{5})^2 $$:
   $$
   (2 - \sqrt{5})^2 = 4 - 4\sqrt{5} + 5 = 9 - 4\sqrt{5}
   $$

2. Now square that result:
   $$
   (9 - 4\sqrt{5})^2 = 81 - 72\sqrt{5} + 80 = 161 - 72\sqrt{5}
   $$

Thus, we have:

$$
f(2 - \sqrt{5}) = (161 - 72\sqrt{5}) - 16 = 145 - 72\sqrt{5}
$$

### Final Results

- $$ f(1) = -15 $$
- $$ f(-1) = -15 $$
- $$ f(2 - \sqrt{5}) = 145 - 72\sqrt{5} $$

These are the function values at the specified points.
$$ f(1) = -15 $$, $$ f(-1) = -15 $$, and $$ f(2 - \sqrt{5}) = 145 - 72\sqrt{5} $$.

\( 1 \leftarrow \quad \) Use synthetic division to find the function values. \( f(x)=x^{4}-16 \); find \( f(1), f(-1) \), and \( f(2-\sqrt{5}) \).

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