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Thread: 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 $\displaystyle (3x\pm k)$, where 'k' refers to any of the factors of $\displaystyle 2$, by testing $\displaystyle f(\frac{-k}{3})$, 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 $\displaystyle (3x\pm k)$, where 'k' refers to any of the factors of $\displaystyle 2$, by testing $\displaystyle f(\frac{-k}{3})$, 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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