How do you determine the number of real roots in a polynomial?
For a general polynominal, the cubic, quartic, and greater
formulæ are too hellishly hard to work with, so you would need to
plot the function or use Newton's/somesuch method to count the real
roots by hand. If the polynomial has integral roots, you can use
synthetic division to peel off the degrees to see if they factor
wholely into binominals; then all roots will be real and explicit.
Good luck: