Answer :
Let's solve the synthetic division problem step-by-step to find the remainder.
We start with the polynomial [tex]\( f(x) = x^2 - 3x + 122 \)[/tex] and are given [tex]\( k = 3 \)[/tex], which represents the value of [tex]\( x \)[/tex] at which the synthetic division will be performed.
### Step 1: Write down the coefficients
The coefficients of the polynomial [tex]\( f(x) = x^2 - 3x + 122 \)[/tex] are:
[tex]\[ [1, -3, 122] \][/tex]
### Step 2: Set up the synthetic division
We set up the synthetic division using the given [tex]\( k = 3 \)[/tex]:
```
| 3
|----------------
1 | 1 -3 122
```
### Step 3: Perform the synthetic division
1. Bring down the first coefficient (1):
```
| 3
|----------------
1 | 1
```
2. Multiply 3 (the value of [tex]\( k \)[/tex]) by the number below the line (1), and write the result under the next coefficient (-3):
```
| 3
|----------------
1 | 1 3
0
```
3. Add the number just written (3) to the coefficient above it (-3), and write the result below this addition:
```
| 3
|----------------
1 | 1 0
```
4. Multiply 3 (the value of [tex]\( k \)[/tex]) by the number just written (0), and write the result under the next coefficient (122):
```
| 3
|----------------
1 | 1 0 3
0 122
```
5. Add the number just written (0) to the coefficient above it (122), and write the result below this addition:
```
| 3
|----------------
1 | 1 0 122
```
### Step 4: Determine the remainder
The last number written at the end of the row is the remainder. In this case, the remainder is:
[tex]\[ \boxed{122} \][/tex]
Thus, the remainder of this synthetic division problem is:
[tex]\[ 122 \][/tex]
Therefore, the correct answer is [tex]\( 122 \)[/tex]. Since none of the provided options in the question (4, 3, 5, 6) match this result, it appears there may be an error in the provided options. The remainder of the synthetic division problem as calculated is indeed [tex]\( 122 \)[/tex].
We start with the polynomial [tex]\( f(x) = x^2 - 3x + 122 \)[/tex] and are given [tex]\( k = 3 \)[/tex], which represents the value of [tex]\( x \)[/tex] at which the synthetic division will be performed.
### Step 1: Write down the coefficients
The coefficients of the polynomial [tex]\( f(x) = x^2 - 3x + 122 \)[/tex] are:
[tex]\[ [1, -3, 122] \][/tex]
### Step 2: Set up the synthetic division
We set up the synthetic division using the given [tex]\( k = 3 \)[/tex]:
```
| 3
|----------------
1 | 1 -3 122
```
### Step 3: Perform the synthetic division
1. Bring down the first coefficient (1):
```
| 3
|----------------
1 | 1
```
2. Multiply 3 (the value of [tex]\( k \)[/tex]) by the number below the line (1), and write the result under the next coefficient (-3):
```
| 3
|----------------
1 | 1 3
0
```
3. Add the number just written (3) to the coefficient above it (-3), and write the result below this addition:
```
| 3
|----------------
1 | 1 0
```
4. Multiply 3 (the value of [tex]\( k \)[/tex]) by the number just written (0), and write the result under the next coefficient (122):
```
| 3
|----------------
1 | 1 0 3
0 122
```
5. Add the number just written (0) to the coefficient above it (122), and write the result below this addition:
```
| 3
|----------------
1 | 1 0 122
```
### Step 4: Determine the remainder
The last number written at the end of the row is the remainder. In this case, the remainder is:
[tex]\[ \boxed{122} \][/tex]
Thus, the remainder of this synthetic division problem is:
[tex]\[ 122 \][/tex]
Therefore, the correct answer is [tex]\( 122 \)[/tex]. Since none of the provided options in the question (4, 3, 5, 6) match this result, it appears there may be an error in the provided options. The remainder of the synthetic division problem as calculated is indeed [tex]\( 122 \)[/tex].
