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Math Help - Zeros of Polynomials - Finding open interval

  1. #1
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    Zeros of Polynomials - Finding open interval

    Hi, I have a question for the above topic.

    Let p(x) =5x^3 -23x^2 + x + 6 be a polynomial. Then all zeros of p(x) lies in [-M,M]. Find the value of M to 2 d.p.

    My solution
    p(x) = x^3 - (23/5)x^2 + (1/5)x +6/5
    M = 1 + max{|-23/5|, |1/5|,|6/5|}
    = 1 + 23/5 = 5.6

    Here, i am not sure if i should divide the constant 6 by the leading coefficient which is 5. If i don't divide, then M = 1 + 6 = 7.

    So, which is the correct one?
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  2. #2
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    Quote Originally Posted by tottijohn View Post
    Hi, I have a question for the above topic.

    Let p(x) =5x^3 -23x^2 + x + 6 be a polynomial. Then all zeros of p(x) lies in [-M,M]. Find the value of M to 2 d.p.

    My solution
    p(x) = x^3 - (23/5)x^2 + (1/5)x +6/5
    M = 1 + max{|-23/5|, |1/5|,|6/5|}
    = 1 + 23/5 = 5.6

    Here, i am not sure if i should divide the constant 6 by the leading coefficient which is 5. If i don't divide, then M = 1 + 6 = 7.

    So, which is the correct one?
    What you have done is a correct application of the Cauchy bound for the roots of a polynomial, that is it gives 5.6.

    In fact the largest absolute value of a root is about 4.5

    CB
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  3. #3
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    Thank you, CaptainBlack for the prompt reply!
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