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What about a 5th degree polynomial?

-- 5 real zeros? Yep!
        It can cross
5 times -- right?

graph of a 5th degree polynomial with 5 real zeros

-- 4 real and 1 complex? ( 4 + 1 = 5 )
        Nope!  That complex dude would be lonely without his conjugate
        buddy.

-- 3 real, 2 complex?  Sure!

-- 2 real, 3 complex?
        Nope -- complex guys have to be in pairs!

-- 1 real, 4 complex?  Yep!
        The graph might look like this

graph of a 5th degree polynomial with 1 real zero and 4 complex zeros

-- No real zeros, 5 complex?
        Not a chance!
        Two problems with this:

1
 
Odd degree polynomials must have, at least,
1 real zero.
 
21 complex guy would be alone.

 


YOUR TURN:

List all possible zero combinations for a 4th degree polynomial.