anonymous
  • anonymous
Hi everybody! Find all positive integral solutions (x,y) of the polynomial equation: 4x^3+4x^2y-15xy^2-18y^3-12x^2+6xy+36y^2+5x-10y=0
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
any thoughts/ideas?
angela210793
  • angela210793
couldn't u find a longer eq? :P
anonymous
  • anonymous
its a monstrous beast...

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More answers

angela210793
  • angela210793
:P
angela210793
  • angela210793
I think it is smth like (ax-by)^3 bu i can't tell wht r a and b
angela210793
  • angela210793
but*
anonymous
  • anonymous
good try angela..but u never know that will factorize completely to 3 linear factors!... Infact this BEAST can be factored to any one of the following: 1.3 linears. 2.1 linear and 1 conic
anonymous
  • anonymous
(x-2*y)*(2*x-1+3*y)*(2*x-5+3*y)
anonymous
  • anonymous
(x-2y)(2x+3y-5)(2x+3y-1)=0
angela210793
  • angela210793
i'm not planning to deal with this..u do it...:P
anonymous
  • anonymous
i got it first, please medal
anonymous
  • anonymous
u rock cantorset! but we cant use comps!
anonymous
  • anonymous
how do you know i use computer
anonymous
  • anonymous
:)
anonymous
  • anonymous
@contorset: lol, its obvious. :P
angela210793
  • angela210793
hey how did u do tht?
anonymous
  • anonymous
well witout at computer its LUCK..
anonymous
  • anonymous
I didn't do it on my own, wolframalpha
anonymous
  • anonymous
maple
angela210793
  • angela210793
and how do u do it in comp?
anonymous
  • anonymous
haha... if u had a good luck u wud put x=2y and inspect!!!
anonymous
  • anonymous
you can do it in wolframalpha,
anonymous
  • anonymous
then u get (x-2y) and get others by dividing
anonymous
  • anonymous
i should not give out my secrets though
anonymous
  • anonymous
k. but tat doesnt solve the problem..
anonymous
  • anonymous
x=2y ? you mean substitute
anonymous
  • anonymous
@saubhik: You need a VERY good luck for that!
anonymous
  • anonymous
yes...
anonymous
  • anonymous
but see tat doesnt..solve the problem
anonymous
  • anonymous
we have to find positive integer solutions..
angela210793
  • angela210793
woww
anonymous
  • anonymous
but you can pick anything, x = 4y for instance
angela210793
  • angela210793
and i thought u guys were smart :P:P:P
anonymous
  • anonymous
i dont see any logic in this problem
anonymous
  • anonymous
ya cantorset that's truly possible but as i said we need luck and brain-breaking insights...
anonymous
  • anonymous
see..now how to find the solution?
anonymous
  • anonymous
for the first one(x-2y), x = 2I y=I
anonymous
  • anonymous
u can have infinite pairs (2n,n) for a positive integer n since x-2y is a factor...what for the other two?
anonymous
  • anonymous
yes
anonymous
  • anonymous
what for 2x+3y-5=0
anonymous
  • anonymous
its obvious (1,1) but are there others..THIS IS DIOPHANTINE!!
anonymous
  • anonymous
OK lets see, x + 3y/2 =5/2 x=(5-3y)/2 5-3y should be even,
anonymous
  • anonymous
so y is odd.
anonymous
  • anonymous
what about 2x+3y=1? yay...theres no positive integers here in this equation..so this is infertile:P
anonymous
  • anonymous
Why do you think so?
anonymous
  • anonymous
how can u get postive integers (x,y) from 2x+3y<5
anonymous
  • anonymous
there are none...:P
anonymous
  • anonymous
infertile..mother has no babies:P
anonymous
  • anonymous
that can be solved with euclids algorithm
anonymous
  • anonymous
Oh sorry, i dint read that 'positive' part! :P
anonymous
  • anonymous
this mother 2x+3y=5 has only one baby i.e. (1,1)
anonymous
  • anonymous
So our answer: (1,1) and (2n,n)
angela210793
  • angela210793
OMG!!! This is giving me headache only by reading ur explanations imagine if i had to solve this....:@@@@@@@@
anonymous
  • anonymous
So its done, (n,2n) U (1,1)
anonymous
  • anonymous
Openstudy rocks!! thnx evrybody!
anonymous
  • anonymous
Oops (2n,n). yes right
angela210793
  • angela210793
Going to deal with literature bye and have a good time :)
anonymous
  • anonymous
, amog, you wrote x= 1, y = 21?
anonymous
  • anonymous
@angela: buh-bye. @cantorset: It was an 'I' (eye)
anonymous
  • anonymous
for the first one(x-2y), x = 2I y=I
anonymous
  • anonymous
Yes i corrected it, didn't I?
anonymous
  • anonymous
hi cantorset...do u mind givin me an idea how to have a factor to the eqn..
anonymous
  • anonymous
suppose i have the baddest luck and go on inspecting from x=3y;x=4y....
anonymous
  • anonymous
wait, ... start over . why would i think to use x = 3y or anything like that
anonymous
  • anonymous
also its 3 linear, not a conic
anonymous
  • anonymous
or worser x=3y+7 etc..
anonymous
  • anonymous
yeah
anonymous
  • anonymous
you mean because the solution is ( ax + by ) ( dx + cy + ...
anonymous
  • anonymous
See cantormath..the given equation Infact this BEAST can be factored to any one of the following: 1.3 linears. 2.1 linear and 1 conic
anonymous
  • anonymous
now clearly theres no constant term in this equation...so a linear factor (which must be present) must pass through the origin...
anonymous
  • anonymous
and so we start checking by x=y;x=2y;x=3y...
anonymous
  • anonymous
if the question does not mean to torment us we can get the factor in a few trials...
anonymous
  • anonymous
k..thnx evrybody for helping..:)
anonymous
  • anonymous
wait, youre saying in general, a cubic in x and y can be factored into either 3 linears, or 1 linear and 2 conics? , or 2 linears, and a conic?
anonymous
  • anonymous
what do you mean by constant term in the equation
anonymous
  • anonymous
term with no variables attched..
anonymous
  • anonymous
oh right
anonymous
  • anonymous
the final solution is 3 linears, right?
anonymous
  • anonymous
yups...family of 3 straight line
anonymous
  • anonymous
cool, so ok , theres no constant term, and clearly, (0,0) works
anonymous
  • anonymous
yups..try looking for x=y,2y,3y,...here u need luck
anonymous
  • anonymous
so that tells us that x = ay or y = ax must be one of the solutions?
anonymous
  • anonymous
yes..true
anonymous
  • anonymous
but only for a=2
anonymous
  • anonymous
i.e.(2n,n)
anonymous
  • anonymous
right, but thats only if you do the long division< right?
anonymous
  • anonymous
if you know to assume, 2x = y ?
anonymous
  • anonymous
or wait, you know that (0,0) is a solution and (1,1) is a solution, right>
anonymous
  • anonymous
yes...not assume but see if its correct by replacing 2x by y OR replacing y by 2x
anonymous
  • anonymous
(1,1) is known after we get other factors using x-2y
anonymous
  • anonymous
hmmm, yes but i mean is there something that would make us want to to do x = 2y
anonymous
  • anonymous
oh ,
anonymous
  • anonymous
tat is LUCK!
anonymous
  • anonymous
( 1,-1) is another solution then
anonymous
  • anonymous
OR if u knew the problem beforehand OR if u have wolframalpha open on ur browser. No...-1 is not a POSITIVE integer :P
anonymous
  • anonymous
4x^2 + 9y^2 + 12xy -2x -15y +5 =0 How do you solve this btw?
anonymous
  • anonymous
from discriminant criteria for various conics see if its parabola/ellipse/hyperbola..
anonymous
  • anonymous
k. for positive integer solutions...ALL 2-VARIABLE EQUATION DO NOT HAVE positive integer solutions.....such questions are given after testing them..
anonymous
  • anonymous
OK!
anonymous
  • anonymous
amogh u indian?
anonymous
  • anonymous
So here its a parabola
anonymous
  • anonymous
tats what wolframalpha says...
anonymous
  • anonymous
I didn't use wolframalpha for it? Afterall we in India are not allowed to use any calc! :P
anonymous
  • anonymous
And I like it!
anonymous
  • anonymous
in which class r u?
anonymous
  • anonymous
Just went into XII
anonymous
  • anonymous
wer r u frm?
anonymous
  • anonymous
kolkata passed 12
anonymous
  • anonymous
what are the possible shapes of a third degree x,y equation, a conic is second degree,
anonymous
  • anonymous
Cool, did you give the JEE?
anonymous
  • anonymous
may be: 1.family of straight lines.. 2. a strt line and a conic
anonymous
  • anonymous
gave it! but didnt get it:(
anonymous
  • anonymous
hope u get:)
anonymous
  • anonymous
Thanks :) So what plans?
anonymous
  • anonymous
very cool saubhik, didnt know that
anonymous
  • anonymous
dont know ....awaiting for AIEEE and WBJEE and ISI results...
anonymous
  • anonymous
whats AIEEE?
anonymous
  • anonymous
All India Engineering Entrance Examination (in india)
anonymous
  • anonymous
thnx..cantorset...
anonymous
  • anonymous
All the best buddy!
anonymous
  • anonymous
saubhik, i got a problem for ya
anonymous
  • anonymous
mayb ei should post new one
anonymous
  • anonymous
yups...
anonymous
  • anonymous
post new one..

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