Answer :
Sure! Let's go through the synthetic division step-by-step to solve the problem.
We are given the polynomial [tex]\(2x + 7x + 5\)[/tex] and the divisor [tex]\((x + 1)\)[/tex]. To perform synthetic division, we first focus on the divisor root, which is [tex]\(x = -1\)[/tex].
Here is how we perform synthetic division:
1. Write down the coefficients of the polynomial: [tex]\(2, 7, 5\)[/tex].
2. Divide using the root [tex]\(-1\)[/tex].
3. Start the division by bringing down the leading coefficient, [tex]\(2\)[/tex], to start the quotient.
4. Multiply the root [tex]\(-1\)[/tex] by this value [tex]\(2\)[/tex] and write the result under the next coefficient (7).
5. Add the value from the previous step to the next coefficient:
[tex]\[
(-1) \times 2 + 7 = -2 + 7 = 5
\][/tex]
6. Repeat the process for the next coefficient:
Multiply [tex]\(-1\)[/tex] by the last result [tex]\(5\)[/tex]
[tex]\[
(-1) \times 5 = -5
\][/tex]
7. Add this result [tex]\(-5\)[/tex] to the last coefficient (5):
[tex]\[
5 + (-5) = 0
\][/tex]
Thus, the quotient polynomial after performing synthetic division is [tex]\(2x + 5\)[/tex].
Therefore, the quotient in polynomial form is [tex]\(2x + 5\)[/tex], which matches option C.
We are given the polynomial [tex]\(2x + 7x + 5\)[/tex] and the divisor [tex]\((x + 1)\)[/tex]. To perform synthetic division, we first focus on the divisor root, which is [tex]\(x = -1\)[/tex].
Here is how we perform synthetic division:
1. Write down the coefficients of the polynomial: [tex]\(2, 7, 5\)[/tex].
2. Divide using the root [tex]\(-1\)[/tex].
3. Start the division by bringing down the leading coefficient, [tex]\(2\)[/tex], to start the quotient.
4. Multiply the root [tex]\(-1\)[/tex] by this value [tex]\(2\)[/tex] and write the result under the next coefficient (7).
5. Add the value from the previous step to the next coefficient:
[tex]\[
(-1) \times 2 + 7 = -2 + 7 = 5
\][/tex]
6. Repeat the process for the next coefficient:
Multiply [tex]\(-1\)[/tex] by the last result [tex]\(5\)[/tex]
[tex]\[
(-1) \times 5 = -5
\][/tex]
7. Add this result [tex]\(-5\)[/tex] to the last coefficient (5):
[tex]\[
5 + (-5) = 0
\][/tex]
Thus, the quotient polynomial after performing synthetic division is [tex]\(2x + 5\)[/tex].
Therefore, the quotient in polynomial form is [tex]\(2x + 5\)[/tex], which matches option C.
