Answer :
To solve the synthetic division problem and find the quotient in polynomial form, follow these steps:
1. Identify the Polynomial and the Root:
- We have the polynomial [tex]\(2x^2 + 7x + 5\)[/tex].
- The divisor is in the form [tex]\(x + 1\)[/tex], which means we have a root of [tex]\(-1\)[/tex].
2. Set Up Synthetic Division:
- Write down the coefficients of the polynomial: 2, 7, and 5.
- Write the root [tex]\(-1\)[/tex] to the left, outside the synthetic division bracket.
3. Begin the Synthetic Division Process:
- Bring down the first coefficient, which is 2, directly below the line. This is the initial part of the quotient.
4. Calculate the Next Term:
- Multiply the root [tex]\(-1\)[/tex] by the number you just brought down (2), which gives [tex]\(-2\)[/tex].
- Add this to the next coefficient (7). So, [tex]\(-2 + 7 = 5\)[/tex].
- Write 5 below the line as the next part of the quotient.
5. Repeat for the Next Term:
- Multiply the root [tex]\(-1\)[/tex] by the most recent result (5), which gives [tex]\(-5\)[/tex].
- Add this to the last coefficient (5). So, [tex]\(-5 + 5 = 0\)[/tex].
- Write 0, which represents the remainder of the division.
6. Write the Quotient:
- The quotient polynomial is formed from the numbers you wrote below the line, excluding the remainder.
- The quotient is 2 and 5, which corresponds to the polynomial [tex]\(2x + 5\)[/tex].
Thus, the quotient in polynomial form is [tex]\(2x + 5\)[/tex].
The correct option is:
A. [tex]\(2x + 5\)[/tex]
1. Identify the Polynomial and the Root:
- We have the polynomial [tex]\(2x^2 + 7x + 5\)[/tex].
- The divisor is in the form [tex]\(x + 1\)[/tex], which means we have a root of [tex]\(-1\)[/tex].
2. Set Up Synthetic Division:
- Write down the coefficients of the polynomial: 2, 7, and 5.
- Write the root [tex]\(-1\)[/tex] to the left, outside the synthetic division bracket.
3. Begin the Synthetic Division Process:
- Bring down the first coefficient, which is 2, directly below the line. This is the initial part of the quotient.
4. Calculate the Next Term:
- Multiply the root [tex]\(-1\)[/tex] by the number you just brought down (2), which gives [tex]\(-2\)[/tex].
- Add this to the next coefficient (7). So, [tex]\(-2 + 7 = 5\)[/tex].
- Write 5 below the line as the next part of the quotient.
5. Repeat for the Next Term:
- Multiply the root [tex]\(-1\)[/tex] by the most recent result (5), which gives [tex]\(-5\)[/tex].
- Add this to the last coefficient (5). So, [tex]\(-5 + 5 = 0\)[/tex].
- Write 0, which represents the remainder of the division.
6. Write the Quotient:
- The quotient polynomial is formed from the numbers you wrote below the line, excluding the remainder.
- The quotient is 2 and 5, which corresponds to the polynomial [tex]\(2x + 5\)[/tex].
Thus, the quotient in polynomial form is [tex]\(2x + 5\)[/tex].
The correct option is:
A. [tex]\(2x + 5\)[/tex]
