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Topic: Integer Roots (Read 542 times) |
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Barukh
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Integer Roots
« on: Feb 16th, 2005, 3:34am » |
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Find three non-zero integers a, b, c such that the function f(x) = x3 + ax2 + bx + c and its 1-st and 2-nd derivatives have all integer and distinct roots.
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SMQ
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Re: Integer Roots
« Reply #1 on: Feb 18th, 2005, 8:10am » |
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One possible solution (of many) is :: a = -36, b = 285, c = -250 :: giving roots :: 1, 10, and 25 :: for the cubic, :: 5 and 19 :: for the first derivative, and :: 12 :: for the second derivative.
I started by working out formulas for a, b, and c in terms of the six roots, then made the simplifying assumption that there existed a solution where all the roots were positive. With that assumption I then wrote a program to iterate over all possible positive roots of the first derivative :: which uniquely determine the root of the second derivative :: searching for roots of the cubic :: which must sum to three times the root of the second derivative :: which, because of the assumption that all roots are positive is a finite search space. I gave the "smallest" solution above.
--SMQ
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Barukh
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Re: Integer Roots
« Reply #2 on: Feb 19th, 2005, 5:15am » |
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Nicely done, SMQ and a nice start in the forum! 
If you want, you can try to find the approach that is less computer-dependent.
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SMQ
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Re: Integer Roots
« Reply #3 on: Feb 19th, 2005, 6:51am » |
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on Feb 19th, 2005, 5:15am, Barukh wrote:| If you want, you can try to find the approach that is less computer-dependent. |
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Ooh, discrete math and I have a long history of not getting along well.  I did derive some statements about the even/oddness and multiple-of-threeness of various groups of the roots, but it ended up being simpler not to use them in the program. I think for the moment I'll stick to doing continuous and/or infinite problems "by hand" and letting the machine handle discrete finite ones. 
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