Answer :
To solve this synthetic division problem and find the remainder, follow these steps:
1. Identify the Dividend:
The coefficients of the polynomial given are [tex]\(1, 2, -3, 1\)[/tex]. This represents the polynomial [tex]\(1x^3 + 2x^2 - 3x + 1\)[/tex].
2. Identify the Divisor:
In synthetic division, we use the zero of the divisor. Here, we're dividing by [tex]\(x + 2\)[/tex], so we use [tex]\(-2\)[/tex] as the divisor.
3. Setup the Synthetic Division:
Write down the coefficients of the polynomial: [tex]\(1, 2, -3, 1\)[/tex].
4. Perform the Synthetic Division:
- Start by bringing down the first coefficient: [tex]\(1\)[/tex].
- Multiply this number by [tex]\(-2\)[/tex] (the divisor) and write the result under the next coefficient.
- Add this result to the next coefficient: [tex]\(2 + (-2 \times 1) = 2 - 2 = 0\)[/tex].
- Repeat this process:
- Multiply the new result [tex]\(0\)[/tex] by [tex]\(-2\)[/tex] and add to [tex]\(-3\)[/tex]: [tex]\(-3 + (0 \times -2) = -3\)[/tex].
- Multiply the new result [tex]\(-3\)[/tex] by [tex]\(-2\)[/tex] and add to [tex]\(1\)[/tex]: [tex]\(1 + (-3 \times -2) = 1 + 6 = 7\)[/tex].
5. Find the Remainder:
The last number obtained is the remainder of the division.
Given these calculations, the remainder is 7.
Therefore, the correct answer is:
A. 7
1. Identify the Dividend:
The coefficients of the polynomial given are [tex]\(1, 2, -3, 1\)[/tex]. This represents the polynomial [tex]\(1x^3 + 2x^2 - 3x + 1\)[/tex].
2. Identify the Divisor:
In synthetic division, we use the zero of the divisor. Here, we're dividing by [tex]\(x + 2\)[/tex], so we use [tex]\(-2\)[/tex] as the divisor.
3. Setup the Synthetic Division:
Write down the coefficients of the polynomial: [tex]\(1, 2, -3, 1\)[/tex].
4. Perform the Synthetic Division:
- Start by bringing down the first coefficient: [tex]\(1\)[/tex].
- Multiply this number by [tex]\(-2\)[/tex] (the divisor) and write the result under the next coefficient.
- Add this result to the next coefficient: [tex]\(2 + (-2 \times 1) = 2 - 2 = 0\)[/tex].
- Repeat this process:
- Multiply the new result [tex]\(0\)[/tex] by [tex]\(-2\)[/tex] and add to [tex]\(-3\)[/tex]: [tex]\(-3 + (0 \times -2) = -3\)[/tex].
- Multiply the new result [tex]\(-3\)[/tex] by [tex]\(-2\)[/tex] and add to [tex]\(1\)[/tex]: [tex]\(1 + (-3 \times -2) = 1 + 6 = 7\)[/tex].
5. Find the Remainder:
The last number obtained is the remainder of the division.
Given these calculations, the remainder is 7.
Therefore, the correct answer is:
A. 7
