Answer :
To find the remainder of the synthetic division problem, we'll break it down into simple steps:
1. Understand the Polynomial and Divisor:
The coefficients you have are [tex]\(4\)[/tex], [tex]\(6\)[/tex], and [tex]\(-1\)[/tex]. They represent the polynomial [tex]\(4x^2 + 6x - 1\)[/tex].
We are performing synthetic division with the divisor [tex]\(x - 1\)[/tex]. The root for synthetic division is [tex]\(1\)[/tex] because the divisor is in the form [tex]\(x - c\)[/tex], where [tex]\(c = 1\)[/tex].
2. Set Up the Synthetic Division:
- Write down the coefficients: [tex]\(4\)[/tex], [tex]\(6\)[/tex], [tex]\(-1\)[/tex].
- Use the root [tex]\(1\)[/tex] for synthetic division.
3. Perform Synthetic Division:
- Bring down the first coefficient, [tex]\(4\)[/tex].
- Multiply this number by the root (which is [tex]\(1\)[/tex]) and write the result under the next coefficient.
- Add this result to the next coefficient: [tex]\(6 + (4 \times 1) = 10\)[/tex].
- Repeat the process: Multiply [tex]\(10\)[/tex] by [tex]\(1\)[/tex] and write under the next coefficient.
- Add: [tex]\(-1 + (10 \times 1) = 9\)[/tex].
4. The Remainder:
- After completing these steps, the number you end with, [tex]\(9\)[/tex], is the remainder.
Therefore, the remainder of the synthetic division is [tex]\(9\)[/tex].
So, the correct answer is option C: 9.
1. Understand the Polynomial and Divisor:
The coefficients you have are [tex]\(4\)[/tex], [tex]\(6\)[/tex], and [tex]\(-1\)[/tex]. They represent the polynomial [tex]\(4x^2 + 6x - 1\)[/tex].
We are performing synthetic division with the divisor [tex]\(x - 1\)[/tex]. The root for synthetic division is [tex]\(1\)[/tex] because the divisor is in the form [tex]\(x - c\)[/tex], where [tex]\(c = 1\)[/tex].
2. Set Up the Synthetic Division:
- Write down the coefficients: [tex]\(4\)[/tex], [tex]\(6\)[/tex], [tex]\(-1\)[/tex].
- Use the root [tex]\(1\)[/tex] for synthetic division.
3. Perform Synthetic Division:
- Bring down the first coefficient, [tex]\(4\)[/tex].
- Multiply this number by the root (which is [tex]\(1\)[/tex]) and write the result under the next coefficient.
- Add this result to the next coefficient: [tex]\(6 + (4 \times 1) = 10\)[/tex].
- Repeat the process: Multiply [tex]\(10\)[/tex] by [tex]\(1\)[/tex] and write under the next coefficient.
- Add: [tex]\(-1 + (10 \times 1) = 9\)[/tex].
4. The Remainder:
- After completing these steps, the number you end with, [tex]\(9\)[/tex], is the remainder.
Therefore, the remainder of the synthetic division is [tex]\(9\)[/tex].
So, the correct answer is option C: 9.
