Confusing math problems....help?

[phantome]

okay, so I have to solve these math problems but i'm completely lost as of how to start them/finish them XP

Could you guys somehow help?

Here's one-

Show that there are no positive integers n for which n^4+2n^3+2n^2+2n+1 is a perfect square. Are there any positive integers n for which (above equation) is a perfect square? if so, find all such n

how would I go about solving this?


Thank you

 

[Latro]

Wait, what? Show that there are none...and then find all those that are? That's weird. Are you sure the second equation wasn't a slightly different quartic, like, say...n^4+4n^3+6n^2+4n+1 (which does in fact generate MANY perfect squares)?

Anyhow: a place to start is to factor the quartic. It factors fairly nicely by synthetic division (my second guess got me the first factor and my first guess after that got me the second, at which time it stopped being reducible).

If you're still having trouble showing that there are no such integers n once you have the quartic factored, I'll help you further.

Weird question that I never thought about (which is kinda relevant to this problem): is 0 a perfect square? I mean, 0^2=0, so it should be a perfect square...maybe? And what about 1? Is it a perfect square?

 

[phantome]

I have never heard of synthetic division....though there are other ways to factor it. (I'm still in hs and we haven't learned about this yet)

I have always considered zero a perfect square....But supposedly it isn't.
As for one, i'm not quite sure, I'll get back to you on that later

 

[Latro]

I have never heard of synthetic division....though there are other ways to factor it. (I'm still in hs and we haven't learned about this yet)

I have always considered zero a perfect square....But supposedly it isn't.
As for one, i'm not quite sure, I'll get back to you on that later

You haven't done synthetic division? That sucks, that really sucks. Seriously, learn it now, as it is very useful and makes this problem a helluva lot easier:
http://www.purplemath.com/modules/synthdiv.htm

(Hint: good guesses for problems out of textbooks to start with are 1, -1, 2, and -2. If it's not one of those, you should probably graph it or something to give you some guidance on what to do.)

If you haven't done polynomial long division either and aren't supposed to use one of these, I don't really know how to help you, as my solution depends fundamentally on being able to factor the quartic.

Anyhow, here's the way it factored for me, if this helps:
(n+1)^2(n^2+1)
(You could factor this further using complex numbers but that wouldn't get you anywhere).

 

[phantome]

thanks this actually helped a lot, and we can use anything to solve the equation (as long as we explain how we came to that answer)

so once again, this REALLY helped...a LOT...thank you very much=)