Answer :
To solve the synthetic division problem and find the remainder, we will follow these steps:
1. Identify the Polynomial and the Divisor:
- The polynomial is represented by the coefficients: [tex]\([1, 2, -3, 2]\)[/tex].
- The divisor given is [tex]\(1\)[/tex], which means we are dividing by [tex]\(x - 1\)[/tex].
2. Setup for Synthetic Division:
- We write down the coefficients of the polynomial: [tex]\(1, 2, -3, 2\)[/tex].
- The number used in synthetic division is the root of the divisor, [tex]\(c = 1\)[/tex], because our divisor is [tex]\(x - c\)[/tex].
3. Perform Synthetic Division:
- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply this result by the root (1), and add it to the next coefficient (2): [tex]\(1 \times 1 + 2 = 3\)[/tex].
- Repeat: Multiply the result (3) by the root (1) and add to the next coefficient (-3): [tex]\(3 \times 1 + (-3) = 0\)[/tex].
- Repeat: Multiply the result (0) by the root (1) and add to the last coefficient (2): [tex]\(0 \times 1 + 2 = 2\)[/tex].
4. Interpret the Results:
- The values obtained (1, 3, 0) represent the coefficients of the quotient polynomial, but the last number obtained, [tex]\(2\)[/tex], is the remainder.
5. Conclusion:
- The remainder of this synthetic division problem is [tex]\(2\)[/tex].
Therefore, the remainder when [tex]\(x^3 + 2x^2 - 3x + 2\)[/tex] is divided by [tex]\(x - 1\)[/tex] is [tex]\(\boxed{2}\)[/tex].
1. Identify the Polynomial and the Divisor:
- The polynomial is represented by the coefficients: [tex]\([1, 2, -3, 2]\)[/tex].
- The divisor given is [tex]\(1\)[/tex], which means we are dividing by [tex]\(x - 1\)[/tex].
2. Setup for Synthetic Division:
- We write down the coefficients of the polynomial: [tex]\(1, 2, -3, 2\)[/tex].
- The number used in synthetic division is the root of the divisor, [tex]\(c = 1\)[/tex], because our divisor is [tex]\(x - c\)[/tex].
3. Perform Synthetic Division:
- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply this result by the root (1), and add it to the next coefficient (2): [tex]\(1 \times 1 + 2 = 3\)[/tex].
- Repeat: Multiply the result (3) by the root (1) and add to the next coefficient (-3): [tex]\(3 \times 1 + (-3) = 0\)[/tex].
- Repeat: Multiply the result (0) by the root (1) and add to the last coefficient (2): [tex]\(0 \times 1 + 2 = 2\)[/tex].
4. Interpret the Results:
- The values obtained (1, 3, 0) represent the coefficients of the quotient polynomial, but the last number obtained, [tex]\(2\)[/tex], is the remainder.
5. Conclusion:
- The remainder of this synthetic division problem is [tex]\(2\)[/tex].
Therefore, the remainder when [tex]\(x^3 + 2x^2 - 3x + 2\)[/tex] is divided by [tex]\(x - 1\)[/tex] is [tex]\(\boxed{2}\)[/tex].
