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------------------------------------------------ Complete the synthetic division problem below.

[tex]\[ -1 \, \bigg| \, \begin{array}{ccc}
2 & 7 & 5 \\
\end{array} \][/tex]

What is the quotient in polynomial form?

A. [tex]\(x + 5\)[/tex]

B. [tex]\(2x - 5\)[/tex]

C. [tex]\(2x + 5\)[/tex]

D. [tex]\(x - 5\)[/tex]

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.