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Answer :
Sure! Let's work through the synthetic division step-by-step to find the quotient of [tex]\((2x^2 + 7x + 5) \div (x + 1)\)[/tex].
### Step 1: Set Up Synthetic Division
We want to divide the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex]. In synthetic division, we use the root of the divisor, so for [tex]\(x + 1\)[/tex], we will use [tex]\(-1\)[/tex].
### Step 2: Write Down the Coefficients
The coefficients of the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] are [tex]\([2, 7, 5]\)[/tex].
### Step 3: Start Synthetic Division
1. Bring down the leading coefficient 2. This will be the leading term of our quotient.
- Current quotient: [tex]\([2]\)[/tex]
2. Multiply the first coefficient (2) by [tex]\(-1\)[/tex] and add to the next coefficient (7):
- [tex]\(2 \times (-1) = -2\)[/tex]
- [tex]\(7 + (-2) = 5\)[/tex]
- Current quotient: [tex]\([2, 5]\)[/tex]
3. Multiply the new term (5) by [tex]\(-1\)[/tex] and add to the next coefficient (5):
- [tex]\(5 \times (-1) = -5\)[/tex]
- [tex]\(5 + (-5) = 0\)[/tex] (This 0 is the remainder)
### Step 4: Form the Quotient Polynomial
The synthetic division yields the quotient polynomial from the obtained coefficients, which are [tex]\([2, 5]\)[/tex]. This represents the polynomial [tex]\(2x + 5\)[/tex].
### Conclusion
The quotient of the division is [tex]\(2x + 5\)[/tex]. Therefore, the correct answer is:
C. [tex]\(2x + 5\)[/tex]
### Step 1: Set Up Synthetic Division
We want to divide the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex]. In synthetic division, we use the root of the divisor, so for [tex]\(x + 1\)[/tex], we will use [tex]\(-1\)[/tex].
### Step 2: Write Down the Coefficients
The coefficients of the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] are [tex]\([2, 7, 5]\)[/tex].
### Step 3: Start Synthetic Division
1. Bring down the leading coefficient 2. This will be the leading term of our quotient.
- Current quotient: [tex]\([2]\)[/tex]
2. Multiply the first coefficient (2) by [tex]\(-1\)[/tex] and add to the next coefficient (7):
- [tex]\(2 \times (-1) = -2\)[/tex]
- [tex]\(7 + (-2) = 5\)[/tex]
- Current quotient: [tex]\([2, 5]\)[/tex]
3. Multiply the new term (5) by [tex]\(-1\)[/tex] and add to the next coefficient (5):
- [tex]\(5 \times (-1) = -5\)[/tex]
- [tex]\(5 + (-5) = 0\)[/tex] (This 0 is the remainder)
### Step 4: Form the Quotient Polynomial
The synthetic division yields the quotient polynomial from the obtained coefficients, which are [tex]\([2, 5]\)[/tex]. This represents the polynomial [tex]\(2x + 5\)[/tex].
### Conclusion
The quotient of the division is [tex]\(2x + 5\)[/tex]. Therefore, the correct answer is:
C. [tex]\(2x + 5\)[/tex]
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