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Let's wrap this up by putting some ideas together here to get the big picture...

Here's our list of important theorems:

An n degree polynomial has,
at most,
n real zeros.

 

An odd degree polynomial
has, at least, one real zero.

*Remember that the tails shoot off in opposite directions.

general shape for a 5th degree polynomial

An nth degree polynomial has
EXACTLY  n  ZEROS!
(real and complex)

This isn't a theorem -- but, it IS important!

Complex zeros always occur in
conjugate pairs.