Answer :
Let's solve the synthetic division problem step-by-step to determine the quotient in polynomial form when dividing the polynomial by [tex]\(x - 2\)[/tex].
Synthetic Division Steps:
1. Identify Coefficients and Divisor:
- Coefficients of the polynomial: [tex]\(1, 5, -1, 4\)[/tex]
- Since we are dividing by [tex]\(x - 2\)[/tex], the divisor is [tex]\(2\)[/tex].
2. Set Up the Synthetic Division:
- Write the divisor, [tex]\(2\)[/tex], on the left.
- Write the coefficients [tex]\(1, 5, -1, 4\)[/tex] in a row.
3. Perform the Division:
- First, bring down the leading coefficient [tex]\(1\)[/tex] to start the quotient.
- Multiply this [tex]\(1\)[/tex] by [tex]\(2\)[/tex] (the divisor) and write the result under the next coefficient [tex]\(5\)[/tex].
- Add to get the new value: [tex]\(5 + 2 = 7\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(2\)[/tex] and write the result under [tex]\(-1\)[/tex].
- Add to get the new value: [tex]\(-1 + 14 = 13\)[/tex].
- Multiply [tex]\(13\)[/tex] by [tex]\(2\)[/tex] and write the result under the final coefficient [tex]\(4\)[/tex].
- Add to get the remainder: [tex]\(4 + 26 = 30\)[/tex].
4. Interpret the Result:
- The numbers in the quotient row are [tex]\(1, 7, 13\)[/tex], representing the coefficients of the quotient polynomial.
- The resulting quotient polynomial is [tex]\(x^2 + 7x + 13\)[/tex].
- The last number is the remainder, [tex]\(30\)[/tex], but it doesn’t affect the polynomial form of the quotient.
5. Determine the Correct Option:
- We look for the option that matches the quotient [tex]\(x^2 + 7x + 13\)[/tex].
- From the given choices, none directly match the complete polynomial, but since the polynomial division is slightly more straightforward for identification of single degree terms in choices,
the first term was [tex]\(1x\)[/tex] for the main part of the polynomial quotient minus the remainder, as intended by just associating to [tex]\(x\)[/tex] (simplified). Thus, none of the given options, specifically [tex]\(x+7\)[/tex], only fits synthetically with the construction accustomed for one less degree equation through synthesis within calculations.
The closest answer among the options given would typically require a simplification, but neither choice accurately fits the overall solutions presented.
Synthetic Division Steps:
1. Identify Coefficients and Divisor:
- Coefficients of the polynomial: [tex]\(1, 5, -1, 4\)[/tex]
- Since we are dividing by [tex]\(x - 2\)[/tex], the divisor is [tex]\(2\)[/tex].
2. Set Up the Synthetic Division:
- Write the divisor, [tex]\(2\)[/tex], on the left.
- Write the coefficients [tex]\(1, 5, -1, 4\)[/tex] in a row.
3. Perform the Division:
- First, bring down the leading coefficient [tex]\(1\)[/tex] to start the quotient.
- Multiply this [tex]\(1\)[/tex] by [tex]\(2\)[/tex] (the divisor) and write the result under the next coefficient [tex]\(5\)[/tex].
- Add to get the new value: [tex]\(5 + 2 = 7\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(2\)[/tex] and write the result under [tex]\(-1\)[/tex].
- Add to get the new value: [tex]\(-1 + 14 = 13\)[/tex].
- Multiply [tex]\(13\)[/tex] by [tex]\(2\)[/tex] and write the result under the final coefficient [tex]\(4\)[/tex].
- Add to get the remainder: [tex]\(4 + 26 = 30\)[/tex].
4. Interpret the Result:
- The numbers in the quotient row are [tex]\(1, 7, 13\)[/tex], representing the coefficients of the quotient polynomial.
- The resulting quotient polynomial is [tex]\(x^2 + 7x + 13\)[/tex].
- The last number is the remainder, [tex]\(30\)[/tex], but it doesn’t affect the polynomial form of the quotient.
5. Determine the Correct Option:
- We look for the option that matches the quotient [tex]\(x^2 + 7x + 13\)[/tex].
- From the given choices, none directly match the complete polynomial, but since the polynomial division is slightly more straightforward for identification of single degree terms in choices,
the first term was [tex]\(1x\)[/tex] for the main part of the polynomial quotient minus the remainder, as intended by just associating to [tex]\(x\)[/tex] (simplified). Thus, none of the given options, specifically [tex]\(x+7\)[/tex], only fits synthetically with the construction accustomed for one less degree equation through synthesis within calculations.
The closest answer among the options given would typically require a simplification, but neither choice accurately fits the overall solutions presented.
