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Answer :
To solve the synthetic division problem and find the remainder, we need to follow these steps:
1. Identify the Polynomial and Divisor:
We have a polynomial with coefficients [tex]\(1, 2, -3, 2\)[/tex]. This represents the polynomial [tex]\(x^3 + 2x^2 - 3x + 2\)[/tex].
We are dividing this polynomial by [tex]\(x + 1\)[/tex], which means the divisor [tex]\(x - (-1) = x + 1\)[/tex].
2. Set Up for Synthetic Division:
Since we are dividing by [tex]\(x + 1\)[/tex], our divisor is [tex]\(-1\)[/tex]. We use [tex]\(-1\)[/tex] in our synthetic division process.
3. Perform the Synthetic Division:
Write down the coefficients: [tex]\(1, 2, -3, 2\)[/tex].
- Start by bringing down the first coefficient (1) as it is.
- Multiply this number by the divisor [tex]\(-1\)[/tex], and write the result underneath the second coefficient (2).
- Add this result to the second coefficient to get the new value for the next step.
- Repeat this process for each coefficient.
Steps in detail:
- Bring down the 1:
- Write 1 below the line.
- Multiply 1 by [tex]\(-1\)[/tex] and add to 2:
[tex]\[
\text{First step: } 1 \times (-1) + 2 = 1
\][/tex]
- Multiply 1 by [tex]\(-1\)[/tex] and add to -3:
[tex]\[
\text{Second step: } 1 \times (-1) + (-3) = -4
\][/tex]
- Multiply [tex]\(-4\)[/tex] by [tex]\(-1\)[/tex] and add to 2:
[tex]\[
\text{Third step: } (-4) \times (-1) + 2 = 6
\][/tex]
4. Determine the Remainder:
The last number we obtain after processing all the coefficients is the remainder. In this case, the remainder is 6.
Thus, the remainder of the synthetic division is [tex]\(6\)[/tex]. The correct answer is not listed among the given options, but this number is obtained from the process.
1. Identify the Polynomial and Divisor:
We have a polynomial with coefficients [tex]\(1, 2, -3, 2\)[/tex]. This represents the polynomial [tex]\(x^3 + 2x^2 - 3x + 2\)[/tex].
We are dividing this polynomial by [tex]\(x + 1\)[/tex], which means the divisor [tex]\(x - (-1) = x + 1\)[/tex].
2. Set Up for Synthetic Division:
Since we are dividing by [tex]\(x + 1\)[/tex], our divisor is [tex]\(-1\)[/tex]. We use [tex]\(-1\)[/tex] in our synthetic division process.
3. Perform the Synthetic Division:
Write down the coefficients: [tex]\(1, 2, -3, 2\)[/tex].
- Start by bringing down the first coefficient (1) as it is.
- Multiply this number by the divisor [tex]\(-1\)[/tex], and write the result underneath the second coefficient (2).
- Add this result to the second coefficient to get the new value for the next step.
- Repeat this process for each coefficient.
Steps in detail:
- Bring down the 1:
- Write 1 below the line.
- Multiply 1 by [tex]\(-1\)[/tex] and add to 2:
[tex]\[
\text{First step: } 1 \times (-1) + 2 = 1
\][/tex]
- Multiply 1 by [tex]\(-1\)[/tex] and add to -3:
[tex]\[
\text{Second step: } 1 \times (-1) + (-3) = -4
\][/tex]
- Multiply [tex]\(-4\)[/tex] by [tex]\(-1\)[/tex] and add to 2:
[tex]\[
\text{Third step: } (-4) \times (-1) + 2 = 6
\][/tex]
4. Determine the Remainder:
The last number we obtain after processing all the coefficients is the remainder. In this case, the remainder is 6.
Thus, the remainder of the synthetic division is [tex]\(6\)[/tex]. The correct answer is not listed among the given options, but this number is obtained from the process.
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