`((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 \((1+i)^3\)
We can use the binomial expansion or the formula for the cube of a binomial:
\[ (a+b)^3 = a^3 + b^3 + 3ab(a+b) \]
Here, \(a = 1\) and \(b = i\):
\[ (1+i)^3 = 1^3 + i^3 + 3 \cdot 1 \cdot i \cdot (1+i) \]
Calculating each term:
- \(1^3 = 1\)
- \(i^3 = i^2 \cdot i = -1 \cdot i = -i\)
- \(3 \cdot 1 \cdot i \cdot (1+i) = 3i(1+i) = 3i + 3i^2 = 3i - 3\)

Combining these:
\[ (1+i)^3 = 1 - i + 3i - 3 = -2 + 2i \]

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

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

Calculating the denominator:
\[ (2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \]

Calculating the numerator:
\[ (-2 + 2i)(2 - i) = -4 + 2i + 4i - 2i^2 = -4 + 6i + 2 = -2 + 6i \]

### Step 4: Combine the results
Now we have:
\[ \frac{-2 + 6i}{5} \]
This can be separated into real and imaginary parts:
\[ = \frac{-2}{5} + \frac{6}{5}i \]

### Final Answer
Thus, the final result is:
\[ \frac{-2}{5} + \frac{6}{5}i \]