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Answer :
To solve the synthetic division problem, we want to divide a polynomial by [tex]\( x - 2 \)[/tex] using synthetic division. The coefficients provided are from the polynomial [tex]\( 1x^3 + 5x^2 - 1x + 4 \)[/tex].
Here is a step-by-step guide on how to perform synthetic division:
1. Set Up the Synthetic Division:
- Write down the coefficients of the polynomial: [tex]\( 1, 5, -1, \)[/tex] and [tex]\( 4 \)[/tex].
- Place the divisor [tex]\( x - 2 \)[/tex] on the left. Since it's [tex]\( x - 2 \)[/tex], the number we use for division is [tex]\( 2 \)[/tex].
2. Bring Down the Leading Coefficient:
- Bring down the first coefficient, which is [tex]\( 1 \)[/tex].
3. Perform the Division:
- Multiply the number brought down (in this case, [tex]\( 1 \)[/tex]) by the divisor (which is [tex]\( 2 \)[/tex]), and place the result under the next coefficient.
- Add this result to the next coefficient:
- Multiply [tex]\( 1 \times 2 = 2 \)[/tex].
- Add this to the next coefficient: [tex]\( 5 + 2 = 7 \)[/tex].
- Repeat the process with this new value:
- Multiply [tex]\( 7 \times 2 = 14 \)[/tex].
- Add this to the next coefficient: [tex]\( -1 + 14 = 13 \)[/tex].
- Repeat one more time:
- Multiply [tex]\( 13 \times 2 = 26 \)[/tex].
- Add this to the last coefficient: [tex]\( 4 + 26 = 30 \)[/tex].
4. Interpret the Result:
- The numbers we have obtained represent the coefficients of the quotient and the remainder.
- So, the quotient coefficients are [tex]\( 1, 7, \)[/tex] and [tex]\( 13 \)[/tex], and the remainder is [tex]\( 30 \)[/tex].
5. Write the Quotient in Polynomial Form:
- Based on the quotient coefficients [tex]\( 1 \)[/tex] and [tex]\( 7 \)[/tex], the quotient polynomial is [tex]\( x^2 + 7x + 13 \)[/tex].
Since none of the provided options completely match what we calculated, it's possible there was an expectation for interpreting the calculation differently, or a potential error in options. However, based on resolving the coefficients for the synthetic division provided, we confirm the quotient polynomial derived from those steps.
If you need further assistance, please ask!
Here is a step-by-step guide on how to perform synthetic division:
1. Set Up the Synthetic Division:
- Write down the coefficients of the polynomial: [tex]\( 1, 5, -1, \)[/tex] and [tex]\( 4 \)[/tex].
- Place the divisor [tex]\( x - 2 \)[/tex] on the left. Since it's [tex]\( x - 2 \)[/tex], the number we use for division is [tex]\( 2 \)[/tex].
2. Bring Down the Leading Coefficient:
- Bring down the first coefficient, which is [tex]\( 1 \)[/tex].
3. Perform the Division:
- Multiply the number brought down (in this case, [tex]\( 1 \)[/tex]) by the divisor (which is [tex]\( 2 \)[/tex]), and place the result under the next coefficient.
- Add this result to the next coefficient:
- Multiply [tex]\( 1 \times 2 = 2 \)[/tex].
- Add this to the next coefficient: [tex]\( 5 + 2 = 7 \)[/tex].
- Repeat the process with this new value:
- Multiply [tex]\( 7 \times 2 = 14 \)[/tex].
- Add this to the next coefficient: [tex]\( -1 + 14 = 13 \)[/tex].
- Repeat one more time:
- Multiply [tex]\( 13 \times 2 = 26 \)[/tex].
- Add this to the last coefficient: [tex]\( 4 + 26 = 30 \)[/tex].
4. Interpret the Result:
- The numbers we have obtained represent the coefficients of the quotient and the remainder.
- So, the quotient coefficients are [tex]\( 1, 7, \)[/tex] and [tex]\( 13 \)[/tex], and the remainder is [tex]\( 30 \)[/tex].
5. Write the Quotient in Polynomial Form:
- Based on the quotient coefficients [tex]\( 1 \)[/tex] and [tex]\( 7 \)[/tex], the quotient polynomial is [tex]\( x^2 + 7x + 13 \)[/tex].
Since none of the provided options completely match what we calculated, it's possible there was an expectation for interpreting the calculation differently, or a potential error in options. However, based on resolving the coefficients for the synthetic division provided, we confirm the quotient polynomial derived from those steps.
If you need further assistance, please ask!
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