Answer :
Certainly! Let's take a detailed look at how to use synthetic division to find the remainder for the given problem.
We are dividing the polynomial represented by the coefficients [tex]\(4x^2 + 6x - 3\)[/tex] by [tex]\(x - 1\)[/tex].
Step 1: Set up the synthetic division.
- Use the value [tex]\(1\)[/tex] from [tex]\(x - 1\)[/tex]. This is because synthetic division uses the root of the divisor, which in this case is [tex]\(x = 1\)[/tex].
- Write down the coefficients of the polynomial: [tex]\(4, 6, -3\)[/tex].
```
1 | 4 6 -3
```
Step 2: Begin the process.
1. Bring down the first coefficient [tex]\(4\)[/tex] as it is.
```
| 4 6 -3
1 |-------------
| 4
```
2. Multiply the root, [tex]\(1\)[/tex], by the number you just brought down, [tex]\(4\)[/tex], and write the result under the next coefficient:
[tex]\(1 \times 4 = 4\)[/tex]
```
| 4 6 -3
1 | 4
|-------------
| 4
```
3. Add this result to the next coefficient [tex]\(6\)[/tex]:
[tex]\(6 + 4 = 10\)[/tex]
```
| 4 6 -3
1 | 4
|-------------
| 4 10
```
4. Multiply the root, [tex]\(1\)[/tex], by this new value [tex]\(10\)[/tex] and write the result under the next coefficient:
[tex]\(1 \times 10 = 10\)[/tex]
```
| 4 6 -3
1 | 4 10
|-------------
| 4 10
```
5. Add this result to the last coefficient [tex]\(-3\)[/tex]:
[tex]\(-3 + 10 = 7\)[/tex]
```
| 4 6 -3
1 | 4 10
|-------------
| 4 10 7
```
Step 3: Find the remainder.
The last number you obtain in the bottom row is the remainder. In this case, the remainder is [tex]\(7\)[/tex].
Therefore, the correct answer is B. 7.
We are dividing the polynomial represented by the coefficients [tex]\(4x^2 + 6x - 3\)[/tex] by [tex]\(x - 1\)[/tex].
Step 1: Set up the synthetic division.
- Use the value [tex]\(1\)[/tex] from [tex]\(x - 1\)[/tex]. This is because synthetic division uses the root of the divisor, which in this case is [tex]\(x = 1\)[/tex].
- Write down the coefficients of the polynomial: [tex]\(4, 6, -3\)[/tex].
```
1 | 4 6 -3
```
Step 2: Begin the process.
1. Bring down the first coefficient [tex]\(4\)[/tex] as it is.
```
| 4 6 -3
1 |-------------
| 4
```
2. Multiply the root, [tex]\(1\)[/tex], by the number you just brought down, [tex]\(4\)[/tex], and write the result under the next coefficient:
[tex]\(1 \times 4 = 4\)[/tex]
```
| 4 6 -3
1 | 4
|-------------
| 4
```
3. Add this result to the next coefficient [tex]\(6\)[/tex]:
[tex]\(6 + 4 = 10\)[/tex]
```
| 4 6 -3
1 | 4
|-------------
| 4 10
```
4. Multiply the root, [tex]\(1\)[/tex], by this new value [tex]\(10\)[/tex] and write the result under the next coefficient:
[tex]\(1 \times 10 = 10\)[/tex]
```
| 4 6 -3
1 | 4 10
|-------------
| 4 10
```
5. Add this result to the last coefficient [tex]\(-3\)[/tex]:
[tex]\(-3 + 10 = 7\)[/tex]
```
| 4 6 -3
1 | 4 10
|-------------
| 4 10 7
```
Step 3: Find the remainder.
The last number you obtain in the bottom row is the remainder. In this case, the remainder is [tex]\(7\)[/tex].
Therefore, the correct answer is B. 7.
