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September 6th, 2019, 03:08 PM   #1
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Finding the Zeros of This Function

Hi,

Any ideas on how to algebraically find the zeros of this function?

$\displaystyle f(x) = 0.0002x^3 - 0.08x^2 - 5,000$

I can graph the function and see that it has a zero at x = 500, however, how can I analytically solve for that value? Is there a trick to factoring a polynomial like this?
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September 6th, 2019, 07:38 PM   #2
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https://en.wikipedia.org/wiki/Cubic_function

Above describes solution method.
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September 7th, 2019, 06:23 AM   #3
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$\displaystyle f(x) = 0.0002x^3 - 0.08x^2 - 5,000=\frac1{5000}(x^3-400x^2-25000000)$
Then attack it with the factor theorem. If you use the graph too (a perfectly valid approach), it's very easy to find the root at 500, the factor theorem validates it.
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September 7th, 2019, 07:48 AM   #4
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Try the rational root theorem.
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September 12th, 2019, 12:51 PM   #5
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Finding Zeros

Thanks to all for the responses. However, I am still unable to factor:

$\displaystyle f(x) = \frac{1}{5,000}(x^3 -400x^2 + 25,000,000) $

Can anyone show the steps?
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September 12th, 2019, 01:24 PM   #6
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Quote:
Originally Posted by danabner View Post
Thanks to all for the responses. However, I am still unable to factor:

$\displaystyle f(x) = \frac{1}{5,000}(x^3 -400x^2 - 25,000,000) $
Gah! That positive in your last post screwed me up for a while.

According to the rational root theorem any possible rational solution will be of the form $\displaystyle \pm \dfrac{\text{ Some factor of 25000000}}{1}$

The only thing to do here is find all the factors 25000000 and do some plug'n'chug on it.

You can do it by Cardano's method as mathman suggested. It's more satisfying than factoring 25000000 but the method is not for the squeemish.

-Dan

Last edited by topsquark; September 12th, 2019 at 01:28 PM.
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