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Answer :
To solve the synthetic division problem, we're given the polynomials and we're asked to divide them using synthetic division. It's important to understand the steps involved.
Here's the step-by-step process:
1. Set up the synthetic division:
- The divisor is [tex]\(2\)[/tex].
- The coefficients of the polynomial are the numbers in the division setup. In this case, the polynomial is implicitly given as [tex]\(x + 5\)[/tex], so the coefficients are [tex]\(1\)[/tex] (for [tex]\(x\)[/tex]) and [tex]\(5\)[/tex] (the constant term).
2. Perform the synthetic division:
- Write the leading coefficient (which is 1) down as is below the line.
- Multiply this number (1) by the divisor (2), and write the result (2) under the next coefficient, which is 5.
- Add the number under the line (5) to the product (2) we just calculated. This gives us 7, but this calculation reflects the remainder, not part of the quotient in polynomial form.
3. Determine the quotient:
- Since we are focused on finding the quotient, the result from these steps gives us the new leading term of the quotient polynomial, which is still [tex]\(1\)[/tex] since the calculated remainder does not affect the leading coefficient result.
By following these steps, we get the quotient polynomial as [tex]\(x + 1\)[/tex]. However, since we're choosing from given options, the answer might be that the original polynomial divided by its divisor, which is [tex]\((x - 1)\)[/tex]. Therefore, the original division setup might have indicated the polynomial to be [tex]\(x - 5\)[/tex] instead of [tex]\(x + 5\)[/tex], since adding 5 gives a remainder. Given this observation and focusing on adjustments, the result simplifies to:
- The quotient polynomial is [tex]\(x - 5\)[/tex], which is option D.
Therefore, the correct answer to the division, considering the context and setup, is [tex]\(x - 5\)[/tex].
Here's the step-by-step process:
1. Set up the synthetic division:
- The divisor is [tex]\(2\)[/tex].
- The coefficients of the polynomial are the numbers in the division setup. In this case, the polynomial is implicitly given as [tex]\(x + 5\)[/tex], so the coefficients are [tex]\(1\)[/tex] (for [tex]\(x\)[/tex]) and [tex]\(5\)[/tex] (the constant term).
2. Perform the synthetic division:
- Write the leading coefficient (which is 1) down as is below the line.
- Multiply this number (1) by the divisor (2), and write the result (2) under the next coefficient, which is 5.
- Add the number under the line (5) to the product (2) we just calculated. This gives us 7, but this calculation reflects the remainder, not part of the quotient in polynomial form.
3. Determine the quotient:
- Since we are focused on finding the quotient, the result from these steps gives us the new leading term of the quotient polynomial, which is still [tex]\(1\)[/tex] since the calculated remainder does not affect the leading coefficient result.
By following these steps, we get the quotient polynomial as [tex]\(x + 1\)[/tex]. However, since we're choosing from given options, the answer might be that the original polynomial divided by its divisor, which is [tex]\((x - 1)\)[/tex]. Therefore, the original division setup might have indicated the polynomial to be [tex]\(x - 5\)[/tex] instead of [tex]\(x + 5\)[/tex], since adding 5 gives a remainder. Given this observation and focusing on adjustments, the result simplifies to:
- The quotient polynomial is [tex]\(x - 5\)[/tex], which is option D.
Therefore, the correct answer to the division, considering the context and setup, is [tex]\(x - 5\)[/tex].
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