I just have a general question about the two following propositions:
1) Any polynomial of an odd degree has at least one real 0
2) Any polynomial of degree 3 with real coefficients has at least one real 0.
I would prove (1) using IVT by arguing that if An>0 and we pick an x sufficiently large enough, as limx--> infinity=infinity, and if we pick a negative x sufficiently large enough then limx---> -(infinity)=-(infinity). Thus there is an x s.t. P(x)=0, (by IVT since polynomials are continuous.
Now...for (2), couldn't I just make the same argument? or does the fact that we have a poly of degree 3 change things?
Thanks! And sorry for not knowing how to use latex!