Answer :
To complete the synthetic division problem using the divisor [tex]\(-1\)[/tex] and the polynomial with coefficients [tex]\(2, 7, 5\)[/tex], we follow these steps:
1. Set up the problem for synthetic division:
- Write down the coefficients of the polynomial, which are [tex]\(2, 7,\)[/tex] and [tex]\(5\)[/tex].
- Write the divisor [tex]\(c = -1\)[/tex] to the left.
2. Bring down the leading coefficient:
- Start by bringing down the first coefficient, [tex]\(2\)[/tex], as it is. This is the first part of our quotient.
3. Perform the synthetic division:
- Multiply the current result by the divisor [tex]\(-1\)[/tex] and add it to the next coefficient:
- Multiply [tex]\(2\)[/tex] by [tex]\(-1\)[/tex] to get [tex]\(-2\)[/tex].
- Add [tex]\(-2\)[/tex] to the next coefficient [tex]\(7\)[/tex] to get [tex]\(5\)[/tex]. This is the next part of the quotient.
- Repeat this step for the last coefficient:
- Multiply [tex]\(5\)[/tex] by [tex]\(-1\)[/tex] to get [tex]\(-5\)[/tex].
- Add [tex]\(-5\)[/tex] to the next coefficient [tex]\(5\)[/tex] to get [tex]\(0\)[/tex]. This is the remainder.
4. Write the quotient and remainder:
- The coefficients of the quotient are [tex]\(2\)[/tex] and [tex]\(5\)[/tex]. This corresponds to the polynomial [tex]\(2x + 5\)[/tex].
- The remainder is [tex]\(0\)[/tex], indicating that it divides evenly.
So, the quotient in polynomial form is [tex]\(2x + 5\)[/tex], which matches option C.
1. Set up the problem for synthetic division:
- Write down the coefficients of the polynomial, which are [tex]\(2, 7,\)[/tex] and [tex]\(5\)[/tex].
- Write the divisor [tex]\(c = -1\)[/tex] to the left.
2. Bring down the leading coefficient:
- Start by bringing down the first coefficient, [tex]\(2\)[/tex], as it is. This is the first part of our quotient.
3. Perform the synthetic division:
- Multiply the current result by the divisor [tex]\(-1\)[/tex] and add it to the next coefficient:
- Multiply [tex]\(2\)[/tex] by [tex]\(-1\)[/tex] to get [tex]\(-2\)[/tex].
- Add [tex]\(-2\)[/tex] to the next coefficient [tex]\(7\)[/tex] to get [tex]\(5\)[/tex]. This is the next part of the quotient.
- Repeat this step for the last coefficient:
- Multiply [tex]\(5\)[/tex] by [tex]\(-1\)[/tex] to get [tex]\(-5\)[/tex].
- Add [tex]\(-5\)[/tex] to the next coefficient [tex]\(5\)[/tex] to get [tex]\(0\)[/tex]. This is the remainder.
4. Write the quotient and remainder:
- The coefficients of the quotient are [tex]\(2\)[/tex] and [tex]\(5\)[/tex]. This corresponds to the polynomial [tex]\(2x + 5\)[/tex].
- The remainder is [tex]\(0\)[/tex], indicating that it divides evenly.
So, the quotient in polynomial form is [tex]\(2x + 5\)[/tex], which matches option C.
