The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) will have exactly \( n \) roots, counted with multiplicity, in the complex number system.
The polynomial \( f(x) = 4x^2 - 14 + x^8 \) is a polynomial of degree 8 (since the highest power of \( x \) is 8).
Since the graph has 2 x-intercepts, this means there are 2 real roots.
Now, we calculate the total number of roots:
- Total roots (including real and imaginary) = degree of the polynomial = 8.
- Number of real roots = 2.
To find the number of imaginary roots:
- Total roots = 8
- Real roots = 2
Imaginary roots (which occur in conjugate pairs) can be calculated as: \[ \text{Imaginary roots} = \text{Total roots} - \text{Real roots} = 8 - 2 = 6. \]
Since imaginary roots occur in conjugate pairs, this means there are 6 imaginary roots that divide into 3 pairs.
Thus, the number of imaginary roots is 6.