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Math Help - Polynomial Roots

  1. #1
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    Polynomial Roots

    How would you show that the polynomial p(x) = x^4 + 7 x^3-9 has at least two real roots?
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  2. #2
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    p(-8) > 0
    p(0) < 0
    p(2) > 0

    Since any polynomial is continuous, it follows from the intermediate value theorem.
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  3. #3
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    Quote Originally Posted by Black View Post
    p(-8) > 0
    p(0) < 0
    p(2) > 0

    Since any polynomial is continuous, it follows from the intermediate value theorem.
    How does that prove there's at least 2 roots??
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  4. #4
    Member Black's Avatar
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    By the IVT, since p(0) < 0 < p(-8) and p(0) < 0 < p(2), there exist real numbers c in [-8,0] and d in [0,2] such that p(c)= p(d) = 0.
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  5. #5
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    Quote Originally Posted by CrazyCat87 View Post
    How would you show that the polynomial p(x) = x^4 + 7 x^3-9 has at least two real roots?
    Descartes rule of signs show that this has one positive root, it also shows that this has one negative root. Hence it has at least two real roots.

    CB
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