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Math Help - Finding roots of a polynomial

  1. #1
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    Finding roots of a polynomial

    Prove the polynomial x^6 + x^4 +5x^2 + 1 has at least four real roots.
    This is what I have so far:
    1. f(x)=x^6 + x^4 +5x^2 + 1 is continuous because it is a polynomial
    2. f(0)= 1>0
    3. f(-1)= -2<0
    Since 1-3 are satisfied, there exists x in (0,1) such that for f(x)=0
    Using f(-0.5), f(-0.75), f(-0.875) we can find a root.

    My question is: Is there another way of find the roots besides the way I'm doing it since I not only have to find one root, i have to find four?
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  2. #2
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    Quote Originally Posted by summerset353 View Post
    Prove the polynomial x^6 + x^4 +5x^2 + 1 has at least four real roots.
    This is what I have so far:
    1. f(x)=x^6 + x^4 +5x^2 + 1 is continuous because it is a polynomial
    2. f(0)= 1>0
    3. f(-1)= -2<0
    Since 1-3 are satisfied, there exists x in (0,1) such that for f(x)=0
    Using f(-0.5), f(-0.75), f(-0.875) we can find a root.

    My question is: Is there another way of find the roots besides the way I'm doing it since I not only have to find one root, i have to find four?

    First , f(-1)=8\neq -2 , second the polynomial you wrote has no real roots at all since all the powers of x are even and thus its minimal value is f(0)=1 ...

    Tonio
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