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Math Help - How to calculate zeros withour graphing technology?

  1. #1
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    How to calculate zeros withour graphing technology?

    The function f(x) = 3x^4 + 2x^3 - 15x^2 + 12x -2 is not able to factor without graphing technology because you can only find that the zero is x = 1 without graphing technology.

    Is there a method that I am missing for using test points on the cubic function. I used all plus/minus 0.5,1,2.
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  2. #2
    Super Member Quacky's Avatar
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    Quote Originally Posted by Barthayn View Post
    The function f(x) = 3x^4 + 2x^3 - 15x^2 + 12x -2 is not able to factor without graphing technology because you can only find that the zero is x = 1 without graphing technology.

    Is there a method that I am missing for using test points on the cubic function. I used all plus/minus 0.5,1,2.
    I suppose you could use Descartes' rule of signs to find an approximation for the number of roots.

    But you are really asking about the rational roots test, I think, which is a small expansion of the factor theorem.

    You have to consider that some factors might be of the form (3x\pm k), where 'k' refers to any of the factors of 2, by testing f(\frac{-k}{3}), as you would with the factor theorem. You should find that there are indeed more factors to this polynomial.
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  3. #3
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    Quote Originally Posted by Quacky View Post
    I suppose you could use Descartes' rule of signs to find an approximation for the number of roots.

    But you are really asking about the rational roots test, I think, which is a small expansion of the factor theorem.

    You have to consider that some factors might be of the form (3x\pm k), where 'k' refers to any of the factors of 2, by testing f(\frac{-k}{3}), as you would with the factor theorem. You should find that there are indeed more factors to this polynomial.
    I know that there are more factors, however, the zeros are x = -2.89681, x = 0.23013, and x = 1.

    The -2.89 and 0.23 zeros cannot come from any test points that one can find. Therefore, one must use graphing technology, correct?



    EDIT: Nevermind, I forgot that there was a (x-1)^2 on this function. I feel like a mathematical fool.
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