What about a 5th degree polynomial?

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

-- 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

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

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

 

YOUR TURN:

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