Answer :
To find [tex]\( f(-5) \)[/tex] using synthetic division for the polynomial [tex]\( f(x) = 8x^5 + 39x^4 - 3x^3 + 12x^2 + 15x + 25 \)[/tex], follow these steps:
1. Write down the coefficients: For the polynomial [tex]\( f(x) = 8x^5 + 39x^4 - 3x^3 + 12x^2 + 15x + 25 \)[/tex], the coefficients are 8, 39, -3, 12, 15, and 25.
2. Set up the synthetic division: We'll be working with [tex]\( x = -5 \)[/tex]. Start the process by writing -5 on the left and the coefficients in a row:
```
-5 | 8 39 -3 12 15 25
|_______________________
```
3. Bring down the first coefficient: Write the first coefficient, 8, below the line.
```
-5 | 8 39 -3 12 15 25
|
|_______________________
8
```
4. Multiply and add: Multiply -5 by 8 (the number below the line) and write the result under the second coefficient. Then, add the result to that coefficient.
```
-5 | 8 39 -3 12 15 25
| -40
|_______________________
8 -1
```
5. Repeat the process: Continue this process of multiplying and adding for each coefficient.
- Multiply -5 by the last result (-1), add the result to the next coefficient (-3).
- Multiply the new result by -5, add to the next coefficient (12).
- Continue this process all the way to the last coefficient.
```
-5 | 8 39 -3 12 15 25
| -40 5 -10 -10 -50
|_______________________
8 -1 2 2 5 0
```
6. Read the remainder: The last number on the bottom row after completing the synthetic division is the remainder. This remainder is the value of [tex]\( f(-5) \)[/tex].
From this synthetic division process, the remainder is 0. Therefore, [tex]\( f(-5) = 0 \)[/tex].
1. Write down the coefficients: For the polynomial [tex]\( f(x) = 8x^5 + 39x^4 - 3x^3 + 12x^2 + 15x + 25 \)[/tex], the coefficients are 8, 39, -3, 12, 15, and 25.
2. Set up the synthetic division: We'll be working with [tex]\( x = -5 \)[/tex]. Start the process by writing -5 on the left and the coefficients in a row:
```
-5 | 8 39 -3 12 15 25
|_______________________
```
3. Bring down the first coefficient: Write the first coefficient, 8, below the line.
```
-5 | 8 39 -3 12 15 25
|
|_______________________
8
```
4. Multiply and add: Multiply -5 by 8 (the number below the line) and write the result under the second coefficient. Then, add the result to that coefficient.
```
-5 | 8 39 -3 12 15 25
| -40
|_______________________
8 -1
```
5. Repeat the process: Continue this process of multiplying and adding for each coefficient.
- Multiply -5 by the last result (-1), add the result to the next coefficient (-3).
- Multiply the new result by -5, add to the next coefficient (12).
- Continue this process all the way to the last coefficient.
```
-5 | 8 39 -3 12 15 25
| -40 5 -10 -10 -50
|_______________________
8 -1 2 2 5 0
```
6. Read the remainder: The last number on the bottom row after completing the synthetic division is the remainder. This remainder is the value of [tex]\( f(-5) \)[/tex].
From this synthetic division process, the remainder is 0. Therefore, [tex]\( f(-5) = 0 \)[/tex].
