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Answer :
To find the remainder in the synthetic division problem given the division setup, you can use a straightforward method known as synthetic division. Here’s a step-by-step explanation of how synthetic division works using the coefficients provided:
1. Identify the coefficients of the polynomial and the divisor:
- The coefficients of the polynomial are [tex]\(1, 2, -3, 3\)[/tex].
- The divisor is indicated by the first number, which is [tex]\(1\)[/tex], meaning the polynomial is divided by [tex]\(x - 1\)[/tex], where [tex]\(c = 1\)[/tex].
2. Setup the synthetic division:
- Write down the coefficients of the polynomial: [tex]\(1, 2, -3, 3\)[/tex].
- To the left of this row of coefficients, write the value of [tex]\(c\)[/tex] which is [tex]\(1\)[/tex].
3. Perform the synthetic division:
- Bring down the leading coefficient. In this case, bring down [tex]\(1\)[/tex] to start the bottom row.
- Multiply this number by the divisor ([tex]\(1\)[/tex]) and write the result under the next coefficient. Here, [tex]\(1 \times 1 = 1\)[/tex].
- Add this result to the next coefficient: [tex]\(2 + 1 = 3\)[/tex]. Write [tex]\(3\)[/tex] below the line.
- Repeat the process: Multiply [tex]\(3\)[/tex] by [tex]\(1\)[/tex] giving [tex]\(3\)[/tex], and add to the next coefficient [tex]\(-3\)[/tex]: [tex]\(-3 + 3 = 0\)[/tex].
- Multiply [tex]\(0\)[/tex] by [tex]\(1\)[/tex], resulting in [tex]\(0\)[/tex], then add to the last coefficient [tex]\(3\)[/tex]: [tex]\(3 + 0 = 3\)[/tex].
4. Determine the remainder:
- The last number obtained at the bottom of the synthetic division is the remainder. In this problem, the remainder is [tex]\(3\)[/tex].
Therefore, the remainder of the division is [tex]\(\boxed{3}\)[/tex].
1. Identify the coefficients of the polynomial and the divisor:
- The coefficients of the polynomial are [tex]\(1, 2, -3, 3\)[/tex].
- The divisor is indicated by the first number, which is [tex]\(1\)[/tex], meaning the polynomial is divided by [tex]\(x - 1\)[/tex], where [tex]\(c = 1\)[/tex].
2. Setup the synthetic division:
- Write down the coefficients of the polynomial: [tex]\(1, 2, -3, 3\)[/tex].
- To the left of this row of coefficients, write the value of [tex]\(c\)[/tex] which is [tex]\(1\)[/tex].
3. Perform the synthetic division:
- Bring down the leading coefficient. In this case, bring down [tex]\(1\)[/tex] to start the bottom row.
- Multiply this number by the divisor ([tex]\(1\)[/tex]) and write the result under the next coefficient. Here, [tex]\(1 \times 1 = 1\)[/tex].
- Add this result to the next coefficient: [tex]\(2 + 1 = 3\)[/tex]. Write [tex]\(3\)[/tex] below the line.
- Repeat the process: Multiply [tex]\(3\)[/tex] by [tex]\(1\)[/tex] giving [tex]\(3\)[/tex], and add to the next coefficient [tex]\(-3\)[/tex]: [tex]\(-3 + 3 = 0\)[/tex].
- Multiply [tex]\(0\)[/tex] by [tex]\(1\)[/tex], resulting in [tex]\(0\)[/tex], then add to the last coefficient [tex]\(3\)[/tex]: [tex]\(3 + 0 = 3\)[/tex].
4. Determine the remainder:
- The last number obtained at the bottom of the synthetic division is the remainder. In this problem, the remainder is [tex]\(3\)[/tex].
Therefore, the remainder of the division is [tex]\(\boxed{3}\)[/tex].
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