If `alpha ` is a root of `x^(4) + x^(2) - 1 = 0` , the value of `(alpha^(6) + 2 alpha^(4))^(2012)` is

Options

A. 0
B. `-1`
C. 1
D. none of these

Answer

To solve the problem, we need to find the value of \((\alpha^6 + 2\alpha^4)^{2012}\) given that \(\alpha\) is a root of the equation \(x^4 + x^2 - 1 = 0\).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \alpha^4 + \alpha^2 - 1 = 0 \]
From this, we can express \(\alpha^4\) in terms of \(\alpha^2\):
\[ \alpha^4 = 1 - \alpha^2 \]

2. **Substitute \(\alpha^4\) into the expression:**
We need to evaluate \((\alpha^6 + 2\alpha^4)^{2012}\). First, we express \(\alpha^6\) in terms of \(\alpha^2\):
\[ \alpha^6 = \alpha^4 \cdot \alpha^2 = (1 - \alpha^2) \cdot \alpha^2 = \alpha^2 - \alpha^4 \]
Now substitute \(\alpha^4\):
\[ \alpha^6 = \alpha^2 - (1 - \alpha^2) = 2\alpha^2 - 1 \]

3. **Now substitute \(\alpha^6\) and \(\alpha^4\) into the expression:**
\[ \alpha^6 + 2\alpha^4 = (2\alpha^2 - 1) + 2(1 - \alpha^2) \]
Simplifying this:
\[ = 2\alpha^2 - 1 + 2 - 2\alpha^2 = 1 \]

4. **Raise the result to the power of 2012:**
\[ (1)^{2012} = 1 \]

### Final Answer:
Thus, the value of \((\alpha^6 + 2\alpha^4)^{2012}\) is:
\[ \boxed{1} \]