Question

Divide f(x) = 3x ^ 3 - 4x ^ 2 - 5x + 8by(x - 2) to find f(2) You must show thorough, handwritten work. You may only use scratch paper, a writing utensil, and a calculator that does not have a computer algebra system.

Answer 1

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.**

1. Write \( 2 \) to the left and the coefficients to the right:

\[
\begin{array}{r|rrrr}
2 & 3 & -4 & -5 & 8 \\
\end{array}
\]

2. Bring down the leading coefficient \( 3 \):

\[
\begin{array}{r|rrrr}
2 & 3 & -4 & -5 & 8 \\
& & & & \\
& 3 & & & \\
\end{array}
\]

3. Multiply \( 2 \) by \( 3 \) and write it under the next coefficient:

\[
\begin{array}{r|rrrr}
2 & 3 & -4 & -5 & 8 \\
& & 6 & & \\
& 3 & & & \\
\end{array}
\]

4. Add \( -4 + 6 = 2 \):

\[
\begin{array}{r|rrrr}
2 & 3 & -4 & -5 & 8 \\
& & 6 & & \\
& 3 & 2 & & \\
\end{array}
\]

5. Multiply \( 2 \) by \( 2 \) and write it under the next coefficient:

\[
\begin{array}{r|rrrr}
2 & 3 & -4 & -5 & 8 \\
& & 6 & 4 & \\
& 3 & 2 & & \\
\end{array}
\]

6. Add \( -5 + 4 = -1 \):

\[
\begin{array}{r|rrrr}
2 & 3 & -4 & -5 & 8 \\
& & 6 & 4 & \\
& 3 & 2 & -1 & \\
\end{array}
\]

7. Multiply \( 2 \) by \( -1 \) and write it under the last coefficient:

\[
\begin{array}{r|rrrr}
2 & 3 & -4 & -5 & 8 \\
& & 6 & 4 & -2 \\
& 3 & 2 & -1 & \\
\end{array}
\]

8. 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:**

\[
\boxed{6}
\]