Question

Can a cubic equation have 3 complex roots? 3 real roots? ...?

Answer 1

No, a cubic equation can not have three complex roots. This is because it turns twice and one end goes to positive infinity and one end goes to negative infinity. Thus, one of these MUST cross the x-axis at some point, meaning y = 0 and a real root exists.

Yes, a cubic equation can have three real roots if it cuts the x-axis three times.