Apply the Fundamental Theorem of Algebra to find the number of imaginary roots for the polynomial f(x) = 4x^2-14+x^8,
if you know its graph has two x-intercepts.
The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots, counting multiplicity.
In this case, the polynomial f(x) = 4x^2 - 14 + x^8 has a degree of 8 because of the highest power of x, which is x^8.
Since we know from the given information that the graph of f(x) has two x-intercepts, we can conclude that the polynomial has exactly 2 real roots.
By applying the Fundamental Theorem of Algebra, the remaining roots must be complex roots.
Therefore, the number of imaginary roots for the polynomial f(x) = 4x^2 - 14 + x^8 is 8 - 2 = 6.
To apply the Fundamental Theorem of Algebra to find the number of imaginary roots for the polynomial f(x) = 4x^2 - 14 + x^8, we need to use the fact that the graph of the polynomial has two x-intercepts.
The Fundamental Theorem of Algebra states that every nth degree polynomial with complex coefficients has exactly n roots, counting multiplicities. In this case, since the polynomial f(x) = 4x^2 - 14 + x^8 has degree 8, we expect it to have 8 roots.
However, we are given that the graph of the polynomial has only two x-intercepts. This suggests that the remaining 6 roots must be imaginary (non-real) roots.
Therefore, based on the given information, we can conclude that the polynomial f(x) = 4x^2 - 14 + x^8 has 6 imaginary roots.
To apply the Fundamental Theorem of Algebra to find the number of imaginary roots for the polynomial f(x) = 4x^2-14+x^8, we need to understand the relationship between the number of x-intercepts and the number of roots.
The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex coefficients has at least one complex root. In other words, a polynomial of degree n will have exactly n roots, including both real and complex roots. Furthermore, complex roots always occur in conjugate pairs.
In this case, since the graph of the polynomial f(x) has two x-intercepts, this means that the polynomial has two real roots.
To find the number of imaginary roots, we subtract the known number of real roots from the degree of the polynomial. In this case, the polynomial is of degree 8, so we can calculate the number of imaginary roots as follows:
Number of imaginary roots = Degree of polynomial - Number of real roots
= 8 - 2
= 6
Therefore, the polynomial f(x) = 4x^2-14+x^8 has 6 imaginary roots.