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Thread: How to find rational roots for a cubic polynomial?

  1. #1
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    Exclamation How to find rational roots for a cubic polynomial?

    Hi there,

    I want to find the x intercept for the function y= x3-9x2+15x+30. The answer from the graphing calculator is x=-1.133. I am aware of the factor theorem and the rational factor theorem (for an integer other than 1 or 0 as a leading coefficient). But in this case neither of these can work. Is there any algebraic method to find this root or trial and error is the only way? Any help will be greatly appreciated.

    Thanks
    Nikunj
    Last edited by Nikunj; Jan 24th 2018 at 09:11 PM.
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  2. #2
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    Re: How to find rational roots for a cubic polynomial?

    There are some rather gnarly formulas for cubics and quartics
    Thanks from Nikunj
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  3. #3
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    Re: How to find rational roots for a cubic polynomial?

    -1.133 is not an exact root. It's an approximation of an irrational root. If it were a rational root, you would be able to find it via the methods you mention.
    Thanks from Nikunj
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  4. #4
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    Re: How to find rational roots for a cubic polynomial?

    I suspected that but didn't know for sure... Thanks for your time and efforts.
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  5. #5
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    Re: How to find rational roots for a cubic polynomial?

    Man..!! these formalas are scary... ... Thanks for your help and efforts..
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  6. #6
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    Re: How to find rational roots for a cubic polynomial?

    Quote Originally Posted by Nikunj View Post
    Hi there,

    I want to find the x intercept for the function y= x3-9x2+15x+30. The answer from the graphing calculator is x=-1.133. I am aware of the factor theorem and the rational factor theorem (for an integer other than 1 or 0 as a leading coefficient). But in this case neither of these can work. Is there any algebraic method to find this root or trial and error is the only way? Any help will be greatly appreciated.

    Thanks
    Nikunj
    Check out:
    https://en.wikipedia.org/wiki/Eisenstein%27s_criterion

    We have 3 divides 9, 15, and 30, but 3 does not divide 1 and $3^2=9$ does not divide 30. Therefore, by Eisenstein's criterion, this polynomial is irreducible over the rationals (it has no rational solutions).
    Thanks from Nikunj
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  7. #7
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    Re: How to find rational roots for a cubic polynomial?

    Thanks for this very useful piece of information. Appreciate your help.
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  8. #8
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    Re: How to find rational roots for a cubic polynomial?

    Quote Originally Posted by Nikunj View Post
    Thanks for this very useful piece of information. Appreciate your help.
    No problem. One note with Eisenstein's Criterion. It can tell you when there are no rational solutions, but the opposite is not true. It cannot tell you that a polynomial does have rational solutions.
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  9. #9
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    Re: How to find rational roots for a cubic polynomial?

    Got it.. Thanks
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