According to the fundamental theorem of algebra, how many roots exist for the polynomial function? f(x) = 8x7 – x5 + x3+6

Since the above polynomial is of degree 7, 7 roots exist.

If the degree is odd and the polynomial coefficients are real, there will be at least one real root, since complex roots come in conjugates.

For the given problem, there is exactly one real root.  All the rest are complex.