Answer :
Sure! Let's go through the synthetic division step-by-step to find the remainder.
Problem Statement:
We need to find the remainder when dividing the polynomial [tex]\(2x^2 - 3x + 1\)[/tex] by [tex]\(x + 2\)[/tex].
### Steps for Synthetic Division:
Step 1: Set up the problem
- Start with the coefficients of the polynomial: [tex]\(2, -3, 1\)[/tex].
- Use the opposite sign of the number added to [tex]\(x\)[/tex] in the divisor. Since the divisor is [tex]\(x + 2\)[/tex], we use [tex]\(-2\)[/tex].
Step 2: Bring down the first coefficient
- Bring down the first coefficient, which is [tex]\(2\)[/tex]. This is our starting remainder.
Step 3: Multiply and Add
- Multiply the current remainder by the divisor ([tex]\(-2\)[/tex]) and add this result to the next coefficient.
1. [tex]\(2 \times (-2) = -4\)[/tex]
2. Add this result to the next coefficient, [tex]\(-3\)[/tex]: [tex]\(-3 + (-4) = -7\)[/tex].
- Now, repeat this multiplication and addition with the new remainder:
1. [tex]\(-7 \times (-2) = 14\)[/tex]
2. Add this to the next coefficient, [tex]\(1\)[/tex]: [tex]\(1 + 14 = 15\)[/tex].
Step 4: Determine the remainder
- The last value obtained is the remainder of the division.
Thus, the remainder of the synthetic division is [tex]\(15\)[/tex].
Problem Statement:
We need to find the remainder when dividing the polynomial [tex]\(2x^2 - 3x + 1\)[/tex] by [tex]\(x + 2\)[/tex].
### Steps for Synthetic Division:
Step 1: Set up the problem
- Start with the coefficients of the polynomial: [tex]\(2, -3, 1\)[/tex].
- Use the opposite sign of the number added to [tex]\(x\)[/tex] in the divisor. Since the divisor is [tex]\(x + 2\)[/tex], we use [tex]\(-2\)[/tex].
Step 2: Bring down the first coefficient
- Bring down the first coefficient, which is [tex]\(2\)[/tex]. This is our starting remainder.
Step 3: Multiply and Add
- Multiply the current remainder by the divisor ([tex]\(-2\)[/tex]) and add this result to the next coefficient.
1. [tex]\(2 \times (-2) = -4\)[/tex]
2. Add this result to the next coefficient, [tex]\(-3\)[/tex]: [tex]\(-3 + (-4) = -7\)[/tex].
- Now, repeat this multiplication and addition with the new remainder:
1. [tex]\(-7 \times (-2) = 14\)[/tex]
2. Add this to the next coefficient, [tex]\(1\)[/tex]: [tex]\(1 + 14 = 15\)[/tex].
Step 4: Determine the remainder
- The last value obtained is the remainder of the division.
Thus, the remainder of the synthetic division is [tex]\(15\)[/tex].
