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].
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].
