Step 1 :The given polynomial is \(3x^{4}-7x^{2}+8\).
Step 2 :The fundamental theorem of algebra states that every non-constant polynomial equation has at least one root in the complex number system. The number of roots is equal to the degree of the polynomial.
Step 3 :In this case, the degree of the polynomial is 4, so there should be 4 roots. However, these roots may be real or complex, and they may be distinct or repeated.
Step 4 :The roots of the polynomial are \(-6^{3/4}\cos(\frac{\atan(\sqrt{47}/7)}{2})/3 - 6^{3/4}i\sin(\frac{\atan(\sqrt{47}/7)}{2})/3\), \(-6^{3/4}\cos(\frac{\atan(\sqrt{47}/7)}{2})/3 + 6^{3/4}i\sin(\frac{\atan(\sqrt{47}/7)}{2})/3\), \(6^{3/4}\cos(\frac{\atan(\sqrt{47}/7)}{2})/3 - 6^{3/4}i\sin(\frac{\atan(\sqrt{47}/7)}{2})/3\), and \(6^{3/4}\cos(\frac{\atan(\sqrt{47}/7)}{2})/3 + 6^{3/4}i\sin(\frac{\atan(\sqrt{47}/7)}{2})/3\).
Step 5 :Final Answer: There are \(\boxed{4}\) roots in the given polynomial according to the fundamental theorem of algebra.