The quotient and remainder obtained from dividing
`2+7x+7x^2+2x^3` by 1+2x are respectively:

Options

A. `1, x^2+3x-2`
B. `0,x^2+3x+2`
C. `x^2+3x+2,0`
D. `x^2+3x-2,1`

Answer

To find the quotient and remainder when dividing the polynomial \(2 + 7x + 7x^2 + 2x^3\) by \(1 + 2x\), we will perform polynomial long division. Here’s a step-by-step solution:

### Step 1: Arrange the Polynomials
First, we write the dividend \(2 + 7x + 7x^2 + 2x^3\) in descending order of powers of \(x\):
\[ 2x^3 + 7x^2 + 7x + 2 \]
We need to divide this by \(1 + 2x\).

### Step 2: Divide the Leading Terms
We start by dividing the leading term of the dividend by the leading term of the divisor:
\[ \frac{2x^3}{2x} = x^2 \]
This means our first term in the quotient is \(x^2\).

### Step 3: Multiply and Subtract
Next, we multiply \(x^2\) by the entire divisor \(1 + 2x\):
\[ x^2(1 + 2x) = x^2 + 2x^3 \]
Now, we subtract this from the original polynomial:
\[ (2x^3 + 7x^2 + 7x + 2) - (2x^3 + x^2) = (7x^2 - x^2) + 7x + 2 = 6x^2 + 7x + 2 \]

### Step 4: Repeat the Process
Now we repeat the process with the new polynomial \(6x^2 + 7x + 2\):
1. Divide the leading term:
\[ \frac{6x^2}{2x} = 3x \]
So, the next term in the quotient is \(3x\).

2. Multiply and subtract:
\[ 3x(1 + 2x) = 3x + 6x^2 \]
Subtract this from \(6x^2 + 7x + 2\):
\[ (6x^2 + 7x + 2) - (6x^2 + 3x) = (7x - 3x) + 2 = 4x + 2 \]

### Step 5: Continue
Now we continue with \(4x + 2\):
1. Divide the leading term:
\[ \frac{4x}{2x} = 2 \]
So, the next term in the quotient is \(2\).

2. Multiply and subtract:
\[ 2(1 + 2x) = 2 + 4x \]
Subtract this from \(4x + 2\):
\[ (4x + 2) - (4x + 2) = 0 \]

### Final Result
At this point, we have completed the division. The quotient is:
\[ x^2 + 3x + 2 \]
And the remainder is:
\[ 0 \]

Thus, the quotient and remainder obtained from dividing \(2 + 7x + 7x^2 + 2x^3\) by \(1 + 2x\) are:
- **Quotient**: \(x^2 + 3x + 2\)
- **Remainder**: \(0\)