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March 5th, 2013, 10:59 PM   #1
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Please help, I cannot figure this out, very short question.

A folium of Descartes is given by x^3 + y^3 = 18xy.

a) Find the integer value of y for the point on the curve when x = 8.

Please answer in steps so that I may know how to do this in the future.
Thank you so much.
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March 5th, 2013, 11:29 PM   #2
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Re: Please help, I cannot figure this out, very short questi

Let and write the resulting cubic in . Next apply the rational roots theorem to find the integral zero. What do you find?
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March 5th, 2013, 11:41 PM   #3
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Re: Please help, I cannot figure this out, very short questi

Quote:
Originally Posted by MarkFL
Let and write the resulting cubic in . Next apply the rational roots theorem to find the integral zero. What do you find?
This would be the formula : 512+ y^3= 144y or y^3-144y+512=0

I tried many ways to go beyond this formula and finding the possible roots for these numbers but this is where I get stuck I cannot find the two roots which would satisfy all the conditions.

Thanks.
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March 5th, 2013, 11:47 PM   #4
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Re: Please Help, I cannot figure this out, very short questi

There are 3 roots, but only 1 is an integer. The rational roots theorem tells us it will come from the list:



So, let

Now, try values from the above list, until you find a value for such that .
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March 5th, 2013, 11:49 PM   #5
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Re: Please help, I cannot figure this out, very short questi

@MarkFL : I doubt if the asker knows Rational root test.

The best way is to iterate and you'll find an integral root. Then use synthetic division to find the quadratic and use quadratic formula to see if there are any other roots.
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March 5th, 2013, 11:58 PM   #6
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Re: Please help, I cannot figure this out, very short questi

Quote:
Originally Posted by MarkFL
There are 3 roots, but only 1 is an integer. The rational roots theorem tells us it will come from the list:



So, let

Now, try values from the above list, until you find a value for such that .

I get it now, thank you Mark
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March 6th, 2013, 05:56 AM   #7
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Re: Please help, I cannot figure this out, very short questi

Substituting x = 8 then dividing y - 144y + 512 by y - 4 gives y - 4y - 128, which has no integer roots.
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March 6th, 2013, 07:49 AM   #8
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Quote:
Originally Posted by mathbalarka
The best way is to iterate and you'll find an integral root. Then use synthetic division . . .
Iteration would be a bit lengthy and one would still need to check the suspected answer by substitution. The iteration may fail to converge to the integer root if the initial guess is too far from it (you don't state how the initial guess would be chosen). Also, the iteration may diverge, but come close to some irrelevant integer value while doing so. If the rational roots theorem isn't known, it's not particularly likely that synthetic division will be known (and there's no obvious reason why that is the best division method).
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