`((1+i)^(3))/(2+i)` is equal to

Options

A. `2/5 - 6/5 i`
B. 0
C. `-1/5 + 5/6 i`
D. `- 2/5 + 6/5i`

Answer

To solve the expression \(\frac{(1+i)^3}{2+i}\), we will follow these steps:

### Step 1: Calculate \((1+i)^3\)
Using the binomial theorem, we can expand \((a+b)^3\):
\[ (1+i)^3 = 1^3 + 3 \cdot 1^2 \cdot i + 3 \cdot 1 \cdot i^2 + i^3 \]
Calculating each term:
- \(1^3 = 1\)
- \(3 \cdot 1^2 \cdot i = 3i\)
- \(3 \cdot 1 \cdot i^2 = 3 \cdot i^2 = 3 \cdot (-1) = -3\)
- \(i^3 = i^2 \cdot i = (-1) \cdot i = -i\)

Now, combine these results:
\[ (1+i)^3 = 1 + 3i - 3 - i = -2 + 2i \]

### Step 2: Substitute back into the expression
Now we can substitute back into the original expression:
\[ \frac{(1+i)^3}{2+i} = \frac{-2 + 2i}{2+i} \]

### Step 3: Multiply by the conjugate of the denominator
To simplify the division, we multiply the numerator and denominator by the conjugate of the denominator, which is \(2-i\):
\[ \frac{-2 + 2i}{2+i} \cdot \frac{2-i}{2-i} = \frac{(-2 + 2i)(2 - i)}{(2+i)(2-i)} \]

### Step 4: Calculate the denominator
The denominator simplifies as follows:
\[ (2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 5 \]

### Step 5: Calculate the numerator
Now, for the numerator:
\[ (-2 + 2i)(2 - i) = -2 \cdot 2 + (-2)(-i) + (2i)(2) + (2i)(-i) \]
Calculating each term:
- \(-2 \cdot 2 = -4\)
- \(-2 \cdot -i = 2i\)
- \(2i \cdot 2 = 4i\)
- \(2i \cdot -i = -2i^2 = -2(-1) = 2\)

Now combine these results:
\[ -4 + 2i + 4i + 2 = -4 + 2 + 6i = -2 + 6i \]

### Step 6: Combine the results
Now we can write the full expression:
\[ \frac{-2 + 6i}{5} = -\frac{2}{5} + \frac{6}{5}i \]

### Final Result
Thus, the expression \(\frac{(1+i)^3}{2+i}\) simplifies to:
\[ -\frac{2}{5} + \frac{6}{5}i \]