If `z_(1)` and `z_(2)` are the complex roots of the equation `(x-3)^(3) + 1=0`, then `z_(1) +z_(2)` equal to

Options

A. 1
B. 3
C. 5
D. 7

Explanation

To solve the equation \((x - 3)^3 + 1 = 0\) and find the sum of the complex roots \(z_1 + z_2\), we can follow these steps:

### Step 1: Rewrite the equation
First, we rewrite the equation:
\[ (x - 3)^3 + 1 = 0 \]
This can be rearranged to:
\[ (x - 3)^3 = -1 \]

### Step 2: Take the cube root
Next, we take the cube root of both sides:
\[ x - 3 = \sqrt[3]{-1} \]
The cube roots of \(-1\) are:
\[ \sqrt[3]{-1} = -1, \quad \omega, \quad \omega^2 \]
where \(\omega = e^{2\pi i / 3}\) is a primitive cube root of unity, and \(\omega^2 = e^{-2\pi i / 3}\).

### Step 3: Solve for \(x\)
Now, we can express \(x\) in terms of these roots:
\[ x - 3 = -1 \implies x = 2 \]
\[ x - 3 = \omega \implies x = 3 + \omega \]
\[ x - 3 = \omega^2 \implies x = 3 + \omega^2 \]

### Step 4: Identify the complex roots
The roots of the equation are:
\[ x_1 = 2, \quad z_1 = 3 + \omega, \quad z_2 = 3 + \omega^2 \]
Here, \(z_1\) and \(z_2\) are the complex roots.

### Step 5: Calculate \(z_1 + z_2\)
Now, we need to find \(z_1 + z_2\):
\[ z_1 + z_2 = (3 + \omega) + (3 + \omega^2) = 6 + \omega + \omega^2 \]

### Step 6: Use the property of cube roots of unity
We know from the properties of cube roots of unity that:
\[ 1 + \omega + \omega^2 = 0 \implies \omega + \omega^2 = -1 \]
Thus, we can substitute:
\[ z_1 + z_2 = 6 + (-1) = 5 \]

### Final Answer
Therefore, the sum of the complex roots \(z_1 + z_2\) is:
\[ \boxed{5} \]