Answer :
To find the remainder in the synthetic division problem, follow these steps:
1. Understand the Expression: We are given a polynomial represented by the coefficients [tex]\(12, -3, \)[/tex] and [tex]\(2\)[/tex], and we are dividing by [tex]\(x - 1\)[/tex]. Therefore, the divisor [tex]\(c\)[/tex] is [tex]\(1\)[/tex].
2. Set Up the Synthetic Division:
- Write down the coefficients of the polynomial: [tex]\(12, -3, 2\)[/tex].
- Place the number [tex]\(1\)[/tex] (from [tex]\(x - 1\)[/tex]) to the left outside the synthetic division bracket.
3. Perform Synthetic Division:
- Bring down the first coefficient [tex]\(12\)[/tex] directly below the line.
- Multiply [tex]\(12\)[/tex] by [tex]\(1\)[/tex] (the divisor) to get [tex]\(12\)[/tex], and write it under the next coefficient, [tex]\(-3\)[/tex].
- Add [tex]\(-3\)[/tex] and [tex]\(12\)[/tex] to get [tex]\(9\)[/tex]. Write this result below the line.
- Multiply [tex]\(9\)[/tex] (the result from the previous addition) by [tex]\(1\)[/tex] to get [tex]\(9\)[/tex], and write it under the next coefficient, [tex]\(2\)[/tex].
- Add [tex]\(2\)[/tex] and [tex]\(9\)[/tex] to get [tex]\(11\)[/tex].
4. Identify the Remainder: The final result from the addition is [tex]\(11\)[/tex]. This last number is the remainder when the polynomial is divided by [tex]\(x - 1\)[/tex].
Given these calculations, the remainder in the synthetic division problem is 11.
However, if you need to choose from the given options, it seems that there may be a misunderstanding, as the remainder [tex]\(11\)[/tex] does not match any of the answer choices provided (A. 4, B. 2, C. 3, D. 5). Be sure to verify the details of the question or consult additional resources if needed.
1. Understand the Expression: We are given a polynomial represented by the coefficients [tex]\(12, -3, \)[/tex] and [tex]\(2\)[/tex], and we are dividing by [tex]\(x - 1\)[/tex]. Therefore, the divisor [tex]\(c\)[/tex] is [tex]\(1\)[/tex].
2. Set Up the Synthetic Division:
- Write down the coefficients of the polynomial: [tex]\(12, -3, 2\)[/tex].
- Place the number [tex]\(1\)[/tex] (from [tex]\(x - 1\)[/tex]) to the left outside the synthetic division bracket.
3. Perform Synthetic Division:
- Bring down the first coefficient [tex]\(12\)[/tex] directly below the line.
- Multiply [tex]\(12\)[/tex] by [tex]\(1\)[/tex] (the divisor) to get [tex]\(12\)[/tex], and write it under the next coefficient, [tex]\(-3\)[/tex].
- Add [tex]\(-3\)[/tex] and [tex]\(12\)[/tex] to get [tex]\(9\)[/tex]. Write this result below the line.
- Multiply [tex]\(9\)[/tex] (the result from the previous addition) by [tex]\(1\)[/tex] to get [tex]\(9\)[/tex], and write it under the next coefficient, [tex]\(2\)[/tex].
- Add [tex]\(2\)[/tex] and [tex]\(9\)[/tex] to get [tex]\(11\)[/tex].
4. Identify the Remainder: The final result from the addition is [tex]\(11\)[/tex]. This last number is the remainder when the polynomial is divided by [tex]\(x - 1\)[/tex].
Given these calculations, the remainder in the synthetic division problem is 11.
However, if you need to choose from the given options, it seems that there may be a misunderstanding, as the remainder [tex]\(11\)[/tex] does not match any of the answer choices provided (A. 4, B. 2, C. 3, D. 5). Be sure to verify the details of the question or consult additional resources if needed.
