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Let: $P(x) = a_nx^n + a_{n-1}x^{n-1} + ..... + a_1x + a_0$ where $a_0a_n < 0$
I have to prove that the Polynomial $P(x)$ has at least one positive root
how can I prove it? Any ideas?

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1 Answer 1

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For $x$ very large, $P(x)\approx a_n x^n$ and so has a sign different from that of $P(0)=a_0$ since $a_0a_n<0$. By the intermediate value theorem, there is a positive root.

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    $\begingroup$ well,i almost understand your answer but can you tell me where to use this : a0an < 0 . It is given from the exercise. can you be more specific?. i would be grateful $\endgroup$
    – socrates
    Commented Nov 12, 2013 at 11:35
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    $\begingroup$ $a_0a_n<0$ means that $a_0$ and $a_n$ have different signs. $\endgroup$
    – lhf
    Commented Nov 12, 2013 at 11:35
  • $\begingroup$ +1, I like this way (though it sounds more abstract in nature).. $\endgroup$
    – user87543
    Commented Nov 12, 2013 at 11:39
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    $\begingroup$ awesome help! i completely understand. If there is anyone with another solution he/she is welcome to give it! thanks! $\endgroup$
    – socrates
    Commented Nov 12, 2013 at 11:41

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