calculus question?

Question

(not looking for a detailed solution, just a hint)--A,B are constants. What is the maximum number of distinct roots of a cubic polynomial of the form p(x)=x^3+A^2x+B? Does such a polynomial p always have that many distinct real roots?

Answer 1

The degree of the polynomial tells you how many roots it does has.

In this case, is of third grade so there are three roots whose nature is determined by the values of the coefficients (it could be the three being real, one real and two complex, two real and one complex, the three complex)

Answer 2

I agree for the most part with the previous answer; however, to my knowledge, there can only be an even number of complex roots (involving imaginary numbers) for any polynomial.

The reason for this is that if a polynomial has a complex root, the complex conjugate (e.g. x-iy is the complex conjugate of x+iy) is also a root. Therefore, for a third degree polynomial, if one complex root is known to exist, then its conjugate is also a root. So, there cannot be only one complex root in any polynomial and there cannot be an odd number of complex roots in any polynomial. (please, someone correct me if I am wrong)

I do agree that the number of roots a polynomial has is equal to its degree; although, they may not always be distinct.