To divide \( f(x) = 3x^3 - 4x^2 - 5x + 8 \) by \( (x - 2) \) and find \( f(2) \), we will use synthetic division. Here are the steps you'll need to follow on your scratch paper:
Step 1: Set up for synthetic division.
We will use the root of the divisor, which is \( 2 \) (since \( x - 2 = 0 \) when \( x = 2 \)). Write down the coefficients of \( f(x) \):
\[ 3, -4, -5, 8 \]
Step 2: Perform synthetic division.
- Write \( 2 \) to the left and the coefficients to the right:
\[ \begin{array}{r|rrrr} 2 & 3 & -4 & -5 & 8 \ \end{array} \]
- Bring down the leading coefficient \( 3 \):
\[ \begin{array}{r|rrrr} 2 & 3 & -4 & -5 & 8 \ & & & & \ & 3 & & & \ \end{array} \]
- Multiply \( 2 \) by \( 3 \) and write it under the next coefficient:
\[ \begin{array}{r|rrrr} 2 & 3 & -4 & -5 & 8 \ & & 6 & & \ & 3 & & & \ \end{array} \]
- Add \( -4 + 6 = 2 \):
\[ \begin{array}{r|rrrr} 2 & 3 & -4 & -5 & 8 \ & & 6 & & \ & 3 & 2 & & \ \end{array} \]
- Multiply \( 2 \) by \( 2 \) and write it under the next coefficient:
\[ \begin{array}{r|rrrr} 2 & 3 & -4 & -5 & 8 \ & & 6 & 4 & \ & 3 & 2 & & \ \end{array} \]
- Add \( -5 + 4 = -1 \):
\[ \begin{array}{r|rrrr} 2 & 3 & -4 & -5 & 8 \ & & 6 & 4 & \ & 3 & 2 & -1 & \ \end{array} \]
- Multiply \( 2 \) by \( -1 \) and write it under the last coefficient:
\[ \begin{array}{r|rrrr} 2 & 3 & -4 & -5 & 8 \ & & 6 & 4 & -2 \ & 3 & 2 & -1 & \ \end{array} \]
- Add \( 8 - 2 = 6 \):
\[ \begin{array}{r|rrrr} 2 & 3 & -4 & -5 & 8 \ & & 6 & 4 & -2 \ & 3 & 2 & -1 & 6 \ \end{array} \]
Step 3: Read the result.
After completing the synthetic division, we find that the remainder is \( 6 \).
Thus, \( f(2) = 6 \).
In conclusion:
\[ \boxed{6} \]