How can you quickly determine the number of roots a polynomial will have by looking at the eqation?

Easy way is to look at the term with the highest exponent. The exponent determines the number of roots. Some might be double roots. THough not all of then will be real. The imaganry ones come in pairs

Examples:

x^4 - 4 roots

x^7 - 7 roots

x^3 - 3 roots