`(2x^(2) + 3x + 1) // (x + 1) = ?`

Options

A. (x + 1)
B. (2x + 1)
C. (x + 3)
D. (2x + 3)

Answer

To solve the division of the polynomial \( (2x^2 + 3x + 1) \) by \( (x + 1) \), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We write the expression as:
\[ \frac{2x^2 + 3x + 1}{x + 1} \]

### Step 2: Divide the leading terms
Divide the leading term of the numerator \( 2x^2 \) by the leading term of the denominator \( x \):
\[ \frac{2x^2}{x} = 2x \]

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

### Step 4: Repeat the process
Now, we need to divide the new polynomial \( (x + 1) \) by \( (x + 1) \):
\[ \frac{x + 1}{x + 1} = 1 \]
Multiply \( 1 \) by \( (x + 1) \):
\[ 1(x + 1) = x + 1 \]
Subtract this from the current polynomial:
\[ (x + 1) - (x + 1) = 0 \]

### Step 5: Write the final answer
Since we have reached a remainder of \( 0 \), the final result of the division is:
\[ 2x + 1 \]

Thus, the answer is:
\[ \frac{2x^2 + 3x + 1}{x + 1} = 2x + 1 \]

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