Of course! Let's find the indicated function value [tex]\( f(2) \)[/tex] using synthetic division and the Remainder Theorem.
Given the polynomial [tex]\( f(x) = 3x^3 - 9x^2 + 5x - 4 \)[/tex] and we need to find [tex]\( f(2) \)[/tex].
### Step-by-Step Solution:
1. Setup for Synthetic Division:
- List the coefficients of the polynomial: [tex]\( [3, -9, 5, -4] \)[/tex].
- Set up the synthetic division process with [tex]\( x = 2 \)[/tex].
2. Initial Setup:
- Write down the coefficients in a row.
- Bring down the leading coefficient (the first number) directly below the line.
```
2 | 3 -9 5 -4
|_____________________
3
```
3. Perform Synthetic Division:
- Multiply the number just written below the line by 2 (the value we are substituting in for [tex]\( x \)[/tex]), and write the result under the next coefficient.
- Add this result to the next coefficient and write the sum below the line.
```
2 | 3 -9 5 -4
| 6
|_____________________
3 -3
```
- Repeat this process for each coefficient:
```
2 | 3 -9 5 -4
| 6 -6
|_____________________
3 -3 -1
```
```
2 | 3 -9 5 -4
| 6 -6 -2
|_____________________
3 -3 -1 -6
```
4. Extract Results:
- The numbers below the line are the results of the synthetic division. The last number is the value of the polynomial evaluated at [tex]\( x = 2 \)[/tex].
Thus, the synthetic division process is as follows:
```
2 | 3 -9 5 -4
| 6 -6 -2
| _______________
3 -3 -1 -6
```
5. Conclusion:
- The last value from the division is [tex]\( -6 \)[/tex].
- Therefore, [tex]\( f(2) = -6 \)[/tex].
Using synthetic division and the Remainder Theorem, we find [tex]\( f(2) = -6 \)[/tex].
