The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) will have exactly \( n \) roots (including real and complex roots, counted with multiplicity).
In the given polynomial \( f(x) = 4x^2 - 14 + x^8 \), the degree is determined by the term with the highest power of \( x \), which is \( x^8 \). Hence, the degree of the polynomial is 8.
Since the polynomial is of degree 8, it has a total of 8 roots. We also know that the graph of the polynomial has two x-intercepts. This means that there are 2 real roots.
The total number of roots is given by the degree of the polynomial: \[ \text{Total roots} = 8 \] \[ \text{Real roots} = 2 \]
To find the number of imaginary (or complex) roots, we can use the relationship: \[ \text{Complex roots} = \text{Total roots} - \text{Real roots} \]
Since complex roots come in conjugate pairs (and every complex root counts as two in the root count), the remaining roots after accounting for the 2 real roots must be complex.
Thus, we calculate: \[ \text{Complex roots} = 8 - 2 = 6 \]
Therefore, since complex roots occur in pairs, we can specifically say: \[ \text{Imaginary roots} = \frac{6}{2} = 3 \text{ pairs of complex roots} \]
So the total number of imaginary roots is 6.
Thus, the polynomial \( f(x) = 4x^2 - 14 + x^8 \) has 6 imaginary roots.