\begin{tabular}{|l} I can list the possible number of real and imaginary roots that a polynomial will have. \\ \hline List the possible number or real and imaginary roots of \( 2 x^{6}-3 x^{4}+x^{2}+5 x-8 \) \\ \hline\end{tabular}

When examining the polynomial \( 2x^6 - 3x^4 + x^2 + 5x - 8 \), we first note that it is of degree 6. According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) will have exactly \( n \) roots, counting multiplicities. These roots can be real or complex (imaginary). Therefore, this polynomial can have: - 6 real roots, - 4 real roots and 2 complex conjugate roots, - 2 real roots and 4 complex conjugate roots, or - 0 real roots and 6 complex conjugate roots. In summary, the polynomial could consist of any combination of roots as long as the total adds up to 6.