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Answer :
Sure! To solve the synthetic division problem and find the remainder, follow these steps:
1. Identify the Divisor: The problem involves dividing by the number `-2`.
2. List the Coefficients: Write down the coefficients of the polynomial you are dividing. The polynomial is represented by its coefficients: [tex]\(1, 2, 2, -3, 1\)[/tex].
3. Set Up for Synthetic Division:
- Write the divisor, which is the opposite of what you see in [tex]\(x - (-2)\)[/tex]. So, we use `-2`.
- Write the coefficients of the polynomial in a row: [tex]\([1, 2, 2, -3, 1]\)[/tex].
4. Perform Synthetic Division:
- First, bring down the leading coefficient (1) to the bottom row.
- Multiply this number by the divisor [tex]\(-2\)[/tex] and write the result under the next coefficient.
- Add the value obtained to the second coefficient and write the result below.
- Repeat this process for each coefficient: multiply the result by [tex]\(-2\)[/tex] and add it to the next coefficient.
Let's break it down:
- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply [tex]\(1\)[/tex] by [tex]\(-2\)[/tex]: [tex]\(-2\)[/tex], add this to the next coefficient [tex]\(2\)[/tex]: [tex]\(0\)[/tex].
- Multiply [tex]\(0\)[/tex] by [tex]\(-2\)[/tex]: [tex]\(0\)[/tex], add this to the next coefficient [tex]\(2\)[/tex]: [tex]\(2\)[/tex].
- Multiply [tex]\(2\)[/tex] by [tex]\(-2\)[/tex]: [tex]\(-4\)[/tex], add it to the next coefficient [tex]\(-3\)[/tex]: [tex]\(-7\)[/tex].
- Multiply [tex]\(-7\)[/tex] by [tex]\(-2\)[/tex]: [tex]\(14\)[/tex], add it to the last coefficient [tex]\(1\)[/tex]: [tex]\(15\)[/tex].
5. Identify the Remainder: The last number you obtain after performing these operations is the remainder of the division. In this case, the remainder is [tex]\(15\)[/tex].
Thus, the remainder when you perform the synthetic division is 15, but since none of the given options match this result, let's reassess the problem setup if needed. Typically, the discrepancy here suggests reviewing the interpretation of the problem or potential options.
1. Identify the Divisor: The problem involves dividing by the number `-2`.
2. List the Coefficients: Write down the coefficients of the polynomial you are dividing. The polynomial is represented by its coefficients: [tex]\(1, 2, 2, -3, 1\)[/tex].
3. Set Up for Synthetic Division:
- Write the divisor, which is the opposite of what you see in [tex]\(x - (-2)\)[/tex]. So, we use `-2`.
- Write the coefficients of the polynomial in a row: [tex]\([1, 2, 2, -3, 1]\)[/tex].
4. Perform Synthetic Division:
- First, bring down the leading coefficient (1) to the bottom row.
- Multiply this number by the divisor [tex]\(-2\)[/tex] and write the result under the next coefficient.
- Add the value obtained to the second coefficient and write the result below.
- Repeat this process for each coefficient: multiply the result by [tex]\(-2\)[/tex] and add it to the next coefficient.
Let's break it down:
- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply [tex]\(1\)[/tex] by [tex]\(-2\)[/tex]: [tex]\(-2\)[/tex], add this to the next coefficient [tex]\(2\)[/tex]: [tex]\(0\)[/tex].
- Multiply [tex]\(0\)[/tex] by [tex]\(-2\)[/tex]: [tex]\(0\)[/tex], add this to the next coefficient [tex]\(2\)[/tex]: [tex]\(2\)[/tex].
- Multiply [tex]\(2\)[/tex] by [tex]\(-2\)[/tex]: [tex]\(-4\)[/tex], add it to the next coefficient [tex]\(-3\)[/tex]: [tex]\(-7\)[/tex].
- Multiply [tex]\(-7\)[/tex] by [tex]\(-2\)[/tex]: [tex]\(14\)[/tex], add it to the last coefficient [tex]\(1\)[/tex]: [tex]\(15\)[/tex].
5. Identify the Remainder: The last number you obtain after performing these operations is the remainder of the division. In this case, the remainder is [tex]\(15\)[/tex].
Thus, the remainder when you perform the synthetic division is 15, but since none of the given options match this result, let's reassess the problem setup if needed. Typically, the discrepancy here suggests reviewing the interpretation of the problem or potential options.
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