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Math Help - Intermediate value theorem

  1. #1
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    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!
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  2. #2
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    Quote Originally Posted by jusstjoe View Post
    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)".
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  3. #3
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    Quote Originally Posted by jusstjoe View Post
    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: z^3 -iz^2+z-i has no real roots.

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