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

- anonymous

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- anonymous

any thoughts/ideas?

- angela210793

couldn't u find a longer eq? :P

- anonymous

its a monstrous beast...

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## More answers

- angela210793

:P

- angela210793

I think it is smth like (ax-by)^3 bu i can't tell wht r a and b

- angela210793

but*

- 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

(x-2*y)*(2*x-1+3*y)*(2*x-5+3*y)

- anonymous

(x-2y)(2x+3y-5)(2x+3y-1)=0

- angela210793

i'm not planning to deal with this..u do it...:P

- anonymous

i got it first, please medal

- anonymous

u rock cantorset! but we cant use comps!

- anonymous

how do you know i use computer

- anonymous

:)

- anonymous

@contorset: lol, its obvious. :P

- angela210793

hey how did u do tht?

- anonymous

well witout at computer its LUCK..

- anonymous

I didn't do it on my own, wolframalpha

- anonymous

maple

- angela210793

and how do u do it in comp?

- anonymous

haha... if u had a good luck u wud put x=2y and inspect!!!

- anonymous

you can do it in wolframalpha,

- anonymous

then u get (x-2y) and get others by dividing

- anonymous

i should not give out my secrets though

- anonymous

k. but tat doesnt solve the problem..

- anonymous

x=2y ? you mean substitute

- anonymous

@saubhik: You need a VERY good luck for that!

- anonymous

yes...

- anonymous

but see tat doesnt..solve the problem

- anonymous

we have to find positive integer solutions..

- angela210793

woww

- anonymous

but you can pick anything, x = 4y for instance

- angela210793

and i thought u guys were smart :P:P:P

- anonymous

i dont see any logic in this problem

- anonymous

ya cantorset that's truly possible but as i said we need luck and brain-breaking insights...

- anonymous

see..now how to find the solution?

- anonymous

for the first one(x-2y),
x = 2I
y=I

- 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

yes

- anonymous

what for 2x+3y-5=0

- anonymous

its obvious (1,1) but are there others..THIS IS DIOPHANTINE!!

- anonymous

OK lets see, x + 3y/2 =5/2
x=(5-3y)/2
5-3y should be even,

- anonymous

so y is odd.

- anonymous

what about 2x+3y=1? yay...theres no positive integers here in this equation..so this is infertile:P

- anonymous

Why do you think so?

- anonymous

how can u get postive integers (x,y) from 2x+3y<5

- anonymous

there are none...:P

- anonymous

infertile..mother has no babies:P

- anonymous

that can be solved with euclids algorithm

- anonymous

Oh sorry, i dint read that 'positive' part! :P

- anonymous

this mother 2x+3y=5 has only one baby i.e. (1,1)

- anonymous

So our answer: (1,1) and (2n,n)

- angela210793

OMG!!! This is giving me headache only by reading ur explanations imagine if i had to solve this....:@@@@@@@@

- anonymous

So its done, (n,2n) U (1,1)

- anonymous

Openstudy rocks!! thnx evrybody!

- anonymous

Oops (2n,n).
yes right

- angela210793

Going to deal with literature bye and have a good time :)

- anonymous

, amog, you wrote x= 1, y = 21?

- anonymous

@angela: buh-bye.
@cantorset: It was an 'I' (eye)

- anonymous

for the first one(x-2y), x = 2I y=I

- anonymous

Yes i corrected it, didn't I?

- anonymous

hi cantorset...do u mind givin me an idea how to have a factor to the eqn..

- anonymous

suppose i have the baddest luck and go on inspecting from x=3y;x=4y....

- anonymous

wait, ... start over . why would i think to use x = 3y or anything like that

- anonymous

also its 3 linear, not a conic

- anonymous

or worser x=3y+7 etc..

- anonymous

yeah

- anonymous

you mean because the solution is ( ax + by ) ( dx + cy + ...

- 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

now clearly theres no constant term in this equation...so a linear factor (which must be present) must pass through the origin...

- anonymous

and so we start checking by x=y;x=2y;x=3y...

- anonymous

if the question does not mean to torment us we can get the factor in a few trials...

- anonymous

k..thnx evrybody for helping..:)

- 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

what do you mean by constant term in the equation

- anonymous

term with no variables attched..

- anonymous

oh right

- anonymous

the final solution is 3 linears, right?

- anonymous

yups...family of 3 straight line

- anonymous

cool, so ok , theres no constant term, and clearly, (0,0) works

- anonymous

yups..try looking for x=y,2y,3y,...here u need luck

- anonymous

so that tells us that x = ay or y = ax must be one of the solutions?

- anonymous

yes..true

- anonymous

but only for a=2

- anonymous

i.e.(2n,n)

- anonymous

right, but thats only if you do the long division< right?

- anonymous

if you know to assume, 2x = y ?

- anonymous

or wait, you know that (0,0) is a solution and (1,1) is a solution, right>

- anonymous

yes...not assume but see if its correct by replacing 2x by y OR replacing y by 2x

- anonymous

(1,1) is known after we get other factors using x-2y

- anonymous

hmmm, yes but i mean is there something that would make us want to to do x = 2y

- anonymous

oh ,

- anonymous

tat is LUCK!

- anonymous

( 1,-1) is another solution then

- 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

4x^2 + 9y^2 + 12xy -2x -15y +5 =0
How do you solve this btw?

- anonymous

from discriminant criteria for various conics see if its parabola/ellipse/hyperbola..

- anonymous

k. for positive integer solutions...ALL 2-VARIABLE EQUATION DO NOT HAVE positive integer solutions.....such questions are given after testing them..

- anonymous

OK!

- anonymous

amogh u indian?

- anonymous

So here its a parabola

- anonymous

tats what wolframalpha says...

- anonymous

I didn't use wolframalpha for it? Afterall we in India are not allowed to use any calc! :P

- anonymous

And I like it!

- anonymous

in which class r u?

- anonymous

Just went into XII

- anonymous

wer r u frm?

- anonymous

kolkata passed 12

- anonymous

what are the possible shapes of a third degree x,y equation, a conic is second degree,

- anonymous

Cool, did you give the JEE?

- anonymous

may be:
1.family of straight lines..
2. a strt line and a conic

- anonymous

gave it! but didnt get it:(

- anonymous

hope u get:)

- anonymous

Thanks :) So what plans?

- anonymous

very cool saubhik, didnt know that

- anonymous

dont know ....awaiting for AIEEE and WBJEE and ISI results...

- anonymous

whats AIEEE?

- anonymous

All India Engineering Entrance Examination (in india)

- anonymous

thnx..cantorset...

- anonymous

All the best buddy!

- anonymous

saubhik, i got a problem for ya

- anonymous

mayb ei should post new one

- anonymous

yups...

- anonymous

post new one..

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