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September 6th, 2019, 02:08 PM  #1 
Newbie Joined: Sep 2019 From: Georgia, USA Posts: 2 Thanks: 0  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? 
September 6th, 2019, 06:38 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,823 Thanks: 723  
September 7th, 2019, 05:23 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 Math Focus: Mainly analysis and algebra 
$\displaystyle f(x) = 0.0002x^3  0.08x^2  5,000=\frac1{5000}(x^3400x^225000000)$ 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. 
September 7th, 2019, 06:48 AM  #4 
Senior Member Joined: Oct 2009 Posts: 865 Thanks: 328 
Try the rational root theorem.

September 12th, 2019, 11:51 AM  #5 
Newbie Joined: Sep 2019 From: Georgia, USA Posts: 2 Thanks: 0  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? 
September 12th, 2019, 12:24 PM  #6  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,272 Thanks: 942 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
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 12:28 PM.  

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