College

Complete the synthetic division problem below.

[tex]-1 \longdiv {2 \, 7 \quad 5}[/tex]

What is the quotient in polynomial form?

A. [tex]x + 5[/tex]
B. [tex]x - 5[/tex]
C. [tex]2x - 5[/tex]
D. [tex]2x + 5[/tex]

Answer :

Let's solve the synthetic division problem step by step to determine the quotient.

We're given the polynomial represented by coefficients [tex]\(2x^2 + 7x + 5\)[/tex] and we need to divide it by [tex]\(x + 1\)[/tex].

### Step 1: Setup for Synthetic Division

In synthetic division, we use the root of the divisor. Here, the divisor is [tex]\(x + 1\)[/tex], so the root is [tex]\(-1\)[/tex].

Our polynomial coefficients are: [tex]\(2, 7, 5\)[/tex].

### Step 2: Perform Synthetic Division

1. Bring down the leading coefficient. This is the coefficient of the highest power of [tex]\(x\)[/tex], which is [tex]\(2\)[/tex].

2. Multiply the root [tex]\(-1\)[/tex] by the coefficient you just brought down ([tex]\(2\)[/tex]), which gives [tex]\(-2\)[/tex].

3. Add this to the next coefficient:
[tex]\[
7 + (-2) = 5
\][/tex]

4. Multiply the root [tex]\(-1\)[/tex] by this result ([tex]\(5\)[/tex]), which gives [tex]\(-5\)[/tex].

5. Add this to the last coefficient:
[tex]\[
5 + (-5) = 0
\][/tex]

### Step 3: Interpret the Results

Now, use these numbers to form the quotient and determine the remainder:

- The quotient consists of the first two numbers from your synthetic division process: [tex]\(2, 5\)[/tex]. This forms the polynomial [tex]\(2x + 5\)[/tex].
- The remainder is the last number, which in this case is [tex]\(0\)[/tex].

Thus, the quotient is [tex]\(2x + 5\)[/tex] with a remainder of 0.

### Conclusion

The quotient in polynomial form is:

[tex]\[ \boxed{2x + 5} \][/tex]

So, the correct multiple choice answer is [tex]\( \text{D. } 2x + 5 \)[/tex].