If ` alpha and beta` are the roots of the equation `x^(2) - 2x + 4 = 0` , then what is the value of ` alpha^(3) + beta^(3)` ?

Options

A. 16
B. -16
C. 8
D. -8

Explanation

To find the value of \( \alpha^3 + \beta^3 \) where \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 - 2x + 4 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients of the quadratic equation
The given quadratic equation is:
\[ x^2 - 2x + 4 = 0 \]
Here, the coefficients are:
- \( A = 1 \)
- \( B = -2 \)
- \( C = 4 \)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{B}{A} = -\frac{-2}{1} = 2 \]
- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{C}{A} = \frac{4}{1} = 4 \]

### Step 3: Use the identity for the sum of cubes
The formula for the sum of cubes of two numbers is:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) \]
Substituting the values we found:
- \( \alpha + \beta = 2 \)
- \( \alpha \beta = 4 \)

### Step 4: Calculate \( \alpha^3 + \beta^3 \)
Now substituting into the identity:
\[ \alpha^3 + \beta^3 = (2)^3 - 3(4)(2) \]
Calculating each term:
- \( (2)^3 = 8 \)
- \( 3(4)(2) = 24 \)

So,
\[ \alpha^3 + \beta^3 = 8 - 24 = -16 \]

### Final Answer
Thus, the value of \( \alpha^3 + \beta^3 \) is:
\[ \boxed{-16} \]