Thread: cubic discriminant

Hi guys, need help with this proof, have spent a while on it and getting no where.
I followed the hint but couldn't do anything useful

Suppose that the cubic polynomial x^3 + Ax + B factors into (x-a)(x-b)(x-c)
Prove that 4A^3 + 27B^2 = 0 if and only if two (or more) of a, b, c are the same. (Hint. Multiply out the right-hand side and compare coefficients to relate A and B to a,b,c)

Cheers for the help!

Originally Posted by liedora

Hi guys, need help with this proof, have spent a while on it and getting no where.
I followed the hint but couldn't do anything useful

Suppose that the cubic polynomial x^3 + Ax + B factors into (x-a)(x-b)(x-c)
Prove that 4A^3 + 27B^2 = 0 if and only if two (or more) of a, b, c are the same. (Hint. Multiply out the right-hand side and compare coefficients to relate A and B to a,b,c)

Cheers for the help!

Hence,








Now, what happens if ? ? ? ?

Hi, I got that far and then didn't know where to go, I tried substituting those values into 4A^4 + 27B^2 but I didn't know what to do from there. Or if that is even the right step to take. I must be missing something!