Answer :
To find the remainder in the synthetic division problem, let's go through a step-by-step process of dividing the polynomial by using synthetic division.
The polynomial given is [tex]\(1x^3 + 2x^2 - 3x + 2\)[/tex], and we need to divide it by [tex]\(x - 1\)[/tex].
1. Identify the coefficients: Start with the coefficients of the polynomial, which are [tex]\(1\)[/tex], [tex]\(2\)[/tex], [tex]\(-3\)[/tex], and [tex]\(2\)[/tex].
2. Set up synthetic division: Since we're dividing by [tex]\(x - 1\)[/tex], we use [tex]\(x = 1\)[/tex] for synthetic division.
3. Perform the synthetic division:
- Step 1: Bring down the first coefficient, which is [tex]\(1\)[/tex].
- Step 2: Multiply this value by the divisor, [tex]\(1\)[/tex]. So, [tex]\(1 \times 1 = 1\)[/tex].
- Step 3: Add this result to the next coefficient: [tex]\(2 + 1 = 3\)[/tex].
- Step 4: Multiply the result by the divisor, [tex]\(3 \times 1 = 3\)[/tex].
- Step 5: Add this to the next coefficient: [tex]\(-3 + 3 = 0\)[/tex].
- Step 6: Multiply this result by the divisor, [tex]\(0 \times 1 = 0\)[/tex].
- Step 7: Add this to the last coefficient: [tex]\(2 + 0 = 2\)[/tex].
4. Result: The final value, [tex]\(2\)[/tex], is the remainder.
Therefore, the remainder when dividing the polynomial [tex]\(x^3 + 2x^2 - 3x + 2\)[/tex] by [tex]\(x - 1\)[/tex] is [tex]\(2\)[/tex].
So, the correct answer is D. 2.
The polynomial given is [tex]\(1x^3 + 2x^2 - 3x + 2\)[/tex], and we need to divide it by [tex]\(x - 1\)[/tex].
1. Identify the coefficients: Start with the coefficients of the polynomial, which are [tex]\(1\)[/tex], [tex]\(2\)[/tex], [tex]\(-3\)[/tex], and [tex]\(2\)[/tex].
2. Set up synthetic division: Since we're dividing by [tex]\(x - 1\)[/tex], we use [tex]\(x = 1\)[/tex] for synthetic division.
3. Perform the synthetic division:
- Step 1: Bring down the first coefficient, which is [tex]\(1\)[/tex].
- Step 2: Multiply this value by the divisor, [tex]\(1\)[/tex]. So, [tex]\(1 \times 1 = 1\)[/tex].
- Step 3: Add this result to the next coefficient: [tex]\(2 + 1 = 3\)[/tex].
- Step 4: Multiply the result by the divisor, [tex]\(3 \times 1 = 3\)[/tex].
- Step 5: Add this to the next coefficient: [tex]\(-3 + 3 = 0\)[/tex].
- Step 6: Multiply this result by the divisor, [tex]\(0 \times 1 = 0\)[/tex].
- Step 7: Add this to the last coefficient: [tex]\(2 + 0 = 2\)[/tex].
4. Result: The final value, [tex]\(2\)[/tex], is the remainder.
Therefore, the remainder when dividing the polynomial [tex]\(x^3 + 2x^2 - 3x + 2\)[/tex] by [tex]\(x - 1\)[/tex] is [tex]\(2\)[/tex].
So, the correct answer is D. 2.
