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Math Help - Continuous Function Proof

  1. #1
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    Continuous Function Proof

    Hi,
    Suppose we have a polynomail of degree n:
    p(x) = a0 + a1x + a2x^2 + ... + anx^n
    where n is an odd number and "an" (the constant) not equal 0. Show that this polynomail has a real root. You can assume that polynomails are continuous.

    So it is pretty intuitive that p(x) as x-> inf is positive and as x-> -inf is negative. I just don't know what facts about polynomials to get
    a p(a) = negative number and a p(b) = positive number. Then I know you can use the intermediate value theorem.

    Thanks
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by tbyou87 View Post
    Hi,
    Suppose we have a polynomail of degree n:
    p(x) = a0 + a1x + a2x^2 + ... + anx^n
    where n is an odd number and "an" (the constant) not equal 0. Show that this polynomail has a real root. You can assume that polynomails are continuous.

    So it is pretty intuitive that p(x) as x-> inf is positive and as x-> -inf is negative. I just don't know what facts about polynomials to get
    a p(a) = negative number and a p(b) = positive number. Then I know you can use the intermediate value theorem.

    Thanks
    first you can assume (without loss of generality) that a_n >0. then take the limit as x goes to infinity (this will make the polynomial go to infinity as well), then take the limit as x goes to negative infinity (this causes the polynomial to go to negative infinity as well, since it is an odd polynomial). now you can apply the intermediate value theorem, since clearly - \infty \le 0 \le \infty.
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  3. #3
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    Say a_n>0 without lose of generality. Define the following sequence x_k = a_n k^n + ... + a_1 k + a_0. The claim is that \lim \ x_k = +\infty. To see that you can write x_k = k^n \left( a_n + ... + \frac{a_1}{k^{n-1}}+\frac{a_0}{k^n} \right) the first term k^n \to \infty but the second term \to a_n since a_n>0 it means this limit goes to +\infty. Thus there is a k so that x_n > 1. Similarly define y_k = a_n (-k)^n+...+a_1 (-k)+a_0. And show \lim y_n = -\infty thus there is a k so that y_k < -1. Meaning we can choose an interval large enough so that the polynomial changes from -1 to 1. Which completes the proof.
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