Answer :
To find the remainder in the synthetic division problem, we need to follow these steps:
1. Understand the Problem: We are dividing the polynomial [tex]\(x^2 + 4x + 6\)[/tex] by [tex]\(x - 1\)[/tex].
2. Set up Synthetic Division:
- The divisor is [tex]\(x - 1\)[/tex], so we use [tex]\(1\)[/tex] for synthetic division.
- Write down the coefficients of the polynomial: [tex]\(1, 4, 6\)[/tex].
3. Start the Process:
- Begin by writing the leading coefficient, which is [tex]\(1\)[/tex], in a row.
- Multiply this leading coefficient by [tex]\(1\)[/tex] (the root from the divisor [tex]\(x - 1\)[/tex]) and add it to the next coefficient, [tex]\(4\)[/tex].
4. Carry on the Synthetic Division:
- Multiply the result obtained from the addition by [tex]\(1\)[/tex] (the same root), and then add it to the next coefficient, [tex]\(6\)[/tex].
5. Calculate Step-by-step:
- First, [tex]\(1\)[/tex] as the starting coefficient remains the same.
- [tex]\(1 \times 1 + 4 = 5\)[/tex].
- [tex]\(5 \times 1 + 6 = 11\)[/tex].
6. Identify the Remainder:
- The last number obtained in this process, [tex]\(11\)[/tex], is the remainder.
Therefore, the remainder when dividing the polynomial [tex]\(x^2 + 4x + 6\)[/tex] by [tex]\(x - 1\)[/tex] is [tex]\(\boxed{11}\)[/tex].
1. Understand the Problem: We are dividing the polynomial [tex]\(x^2 + 4x + 6\)[/tex] by [tex]\(x - 1\)[/tex].
2. Set up Synthetic Division:
- The divisor is [tex]\(x - 1\)[/tex], so we use [tex]\(1\)[/tex] for synthetic division.
- Write down the coefficients of the polynomial: [tex]\(1, 4, 6\)[/tex].
3. Start the Process:
- Begin by writing the leading coefficient, which is [tex]\(1\)[/tex], in a row.
- Multiply this leading coefficient by [tex]\(1\)[/tex] (the root from the divisor [tex]\(x - 1\)[/tex]) and add it to the next coefficient, [tex]\(4\)[/tex].
4. Carry on the Synthetic Division:
- Multiply the result obtained from the addition by [tex]\(1\)[/tex] (the same root), and then add it to the next coefficient, [tex]\(6\)[/tex].
5. Calculate Step-by-step:
- First, [tex]\(1\)[/tex] as the starting coefficient remains the same.
- [tex]\(1 \times 1 + 4 = 5\)[/tex].
- [tex]\(5 \times 1 + 6 = 11\)[/tex].
6. Identify the Remainder:
- The last number obtained in this process, [tex]\(11\)[/tex], is the remainder.
Therefore, the remainder when dividing the polynomial [tex]\(x^2 + 4x + 6\)[/tex] by [tex]\(x - 1\)[/tex] is [tex]\(\boxed{11}\)[/tex].
