Answer :
To solve the synthetic division problem and find the remainder, follow these steps:
1. Set Up the Problem: You are given the polynomial coefficients in order: [tex]\(1, 2, 2, -3, 3\)[/tex]. The first number on the left, outside the division signs (often referred to as the divisor), is 1. This represents the value we use in synthetic division.
2. Write Down the Coefficients: List the coefficients of the polynomial: [tex]\(1, 2, 2, -3, 3\)[/tex].
3. Start the Synthetic Division Process:
- Bring Down the First Coefficient: Write down the first coefficient, 1, below the line. This becomes the starting value.
4. Multiply and Add:
- Multiply this number (1 in this case) by the divisor (1), and write the result under the next coefficient.
- Add this result to the next coefficient (2) and write the sum below the line.
- Repeat this multiply and add process for the rest of the coefficients.
Here's the detailed step-by-step:
- Start: 1
- Multiply: [tex]\(1 \times 1 = 1\)[/tex], add to 2: [tex]\(2 + 1 = 3\)[/tex]
- Multiply: [tex]\(3 \times 1 = 3\)[/tex], add to 2: [tex]\(2 + 3 = 5\)[/tex]
- Multiply: [tex]\(5 \times 1 = 5\)[/tex], add to -3: [tex]\(-3 + 5 = 2\)[/tex]
- Multiply: [tex]\(2 \times 1 = 2\)[/tex], add to 3: [tex]\(3 + 2 = 5\)[/tex]
5. Final Result: The last number you get after adding the results is the remainder of the synthetic division.
6. Conclusion: The remainder in the synthetic division problem is 5.
Therefore, the correct answer is [tex]\( \boxed{5} \)[/tex].
1. Set Up the Problem: You are given the polynomial coefficients in order: [tex]\(1, 2, 2, -3, 3\)[/tex]. The first number on the left, outside the division signs (often referred to as the divisor), is 1. This represents the value we use in synthetic division.
2. Write Down the Coefficients: List the coefficients of the polynomial: [tex]\(1, 2, 2, -3, 3\)[/tex].
3. Start the Synthetic Division Process:
- Bring Down the First Coefficient: Write down the first coefficient, 1, below the line. This becomes the starting value.
4. Multiply and Add:
- Multiply this number (1 in this case) by the divisor (1), and write the result under the next coefficient.
- Add this result to the next coefficient (2) and write the sum below the line.
- Repeat this multiply and add process for the rest of the coefficients.
Here's the detailed step-by-step:
- Start: 1
- Multiply: [tex]\(1 \times 1 = 1\)[/tex], add to 2: [tex]\(2 + 1 = 3\)[/tex]
- Multiply: [tex]\(3 \times 1 = 3\)[/tex], add to 2: [tex]\(2 + 3 = 5\)[/tex]
- Multiply: [tex]\(5 \times 1 = 5\)[/tex], add to -3: [tex]\(-3 + 5 = 2\)[/tex]
- Multiply: [tex]\(2 \times 1 = 2\)[/tex], add to 3: [tex]\(3 + 2 = 5\)[/tex]
5. Final Result: The last number you get after adding the results is the remainder of the synthetic division.
6. Conclusion: The remainder in the synthetic division problem is 5.
Therefore, the correct answer is [tex]\( \boxed{5} \)[/tex].
