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complex coefficients

username5131562

What can you say about what the roots of cubic with complex coefficients will be ?

Reply 1

RDKGames

Study Forum Helper

Original post by LElizabeth02

What can you say about what the roots of cubic with complex coefficients will be ?

They will not necessarily occur in conjugate pairs.

(edited 3 years ago)

Reply 2

DFranklin

Original post by RDKGames

They will not occur in conjugate pairs.

Some of this is going to be a bit pedantic, but I think this question is actually a bit of a minefield.

Firstly,

\mathbb{R} \subset \mathbb{C}

, so arguably

x^2 + 1

is still a poly with complex coefficients.

But even if we exclude this bit of pedantry, obviously

ix^2 + i

is a poly with complex coefficients.

To get around that, let's insist that our poly is monic. Then what about

x^3 - ix^2 + x - i = (x-i)^2(x+i)

. At a minimum, this has (a pair of) roots that come in conjugate pairs; arguably, *all* of its roots come in conjugate pairs (if you say that the only *roots* are x = i, x = -i).

It's probably one of those questions where you really need to ask the OP why they're asking.