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Answer :
To solve the synthetic division problem, you start by dividing the polynomial represented by the coefficients [tex]\(1, 5, -1, 4\)[/tex] by [tex]\(x - 2\)[/tex].
### Steps for Synthetic Division:
1. Identify the Zero of the Divisor:
- We are dividing by [tex]\(x - 2\)[/tex], so the zero is [tex]\(x = 2\)[/tex].
2. Setup the Synthetic Division:
- Write the coefficients of the polynomial: [tex]\([1, 5, -1, 4]\)[/tex].
3. Perform the Synthetic Division:
- First, bring down the leading coefficient (1) directly to the bottom row.
- Multiply this number by the zero of the divisor (2), and write the result under the next coefficient (5).
- Add these two numbers to get the next number in the bottom row (1 [tex]\(\times\)[/tex] 2 = 2; 5 + 2 = 7).
- Repeat this process: multiply 7 by 2, write under the next coefficient (-1), and add (-1 + 14 = 13).
- Multiply 13 by 2, write under the next coefficient (4), and add (4 + 26 = 30).
4. Interpret the Result:
- The numbers in the bottom row, excluding the last one, represent the coefficients of the quotient.
- Therefore, the coefficients [tex]\([1, 7, 13]\)[/tex] give us the quotient [tex]\(x+7\)[/tex].
- The last number in the bottom row (30) is the remainder.
### Conclusion:
The quotient of the polynomial division is the polynomial [tex]\(x + 7\)[/tex]. Therefore, the correct answer is:
D. [tex]\(x+7\)[/tex]
### Steps for Synthetic Division:
1. Identify the Zero of the Divisor:
- We are dividing by [tex]\(x - 2\)[/tex], so the zero is [tex]\(x = 2\)[/tex].
2. Setup the Synthetic Division:
- Write the coefficients of the polynomial: [tex]\([1, 5, -1, 4]\)[/tex].
3. Perform the Synthetic Division:
- First, bring down the leading coefficient (1) directly to the bottom row.
- Multiply this number by the zero of the divisor (2), and write the result under the next coefficient (5).
- Add these two numbers to get the next number in the bottom row (1 [tex]\(\times\)[/tex] 2 = 2; 5 + 2 = 7).
- Repeat this process: multiply 7 by 2, write under the next coefficient (-1), and add (-1 + 14 = 13).
- Multiply 13 by 2, write under the next coefficient (4), and add (4 + 26 = 30).
4. Interpret the Result:
- The numbers in the bottom row, excluding the last one, represent the coefficients of the quotient.
- Therefore, the coefficients [tex]\([1, 7, 13]\)[/tex] give us the quotient [tex]\(x+7\)[/tex].
- The last number in the bottom row (30) is the remainder.
### Conclusion:
The quotient of the polynomial division is the polynomial [tex]\(x + 7\)[/tex]. Therefore, the correct answer is:
D. [tex]\(x+7\)[/tex]
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