# Intermediate value theorem

• May 3rd 2009, 10:56 AM
jusstjoe
Intermediate value theorem
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!
• May 3rd 2009, 11:43 AM
HallsofIvy
Quote:

Originally Posted by jusstjoe
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!

It doesn't change anything, it's just a "special case". And you don't have to "make the same argument" again- just say "since 3 is an odd number, the result follows from (1)".
• May 3rd 2009, 12:20 PM
Plato
Quote:

Originally Posted by jusstjoe
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.

PROBLEM: #1 is not true: \$\displaystyle z^3 -iz^2+z-i\$ has no real roots.

That is why the condition 'with real coefficients' is added to the second one.