Mathematics - 2012-12-02 (page 5 of 7)

The easiest is just to make a drawing and try some integer values.
IF a polynomial has integer solutions this will always be divisible by the constant term!!!

Peter Sheldrick

yup, the constant term is just the product of the roots

N3buchadnezzar

So the only possible integer solutions to this polynomial are $-3,-1,1,3$

Jordan

why?

Old John

9:10 PM

@N3buchadnezzar to be precise, that is only true for monic polynomials :)

Charlie

@nimza hi Alexey!

Peter Sheldrick

(x+a)(x+b)(x+c)...=x^n+...+a*b*c...

Nimza

good to see you @Charlie!

Peter Sheldrick

Elementary symmetric polynomial

In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial P can be expressed as a polynomial in elementary symmetric polynomials: P can be given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree d in n variables for any d ≤ n, and it is formed by adding together all distinct products of d distinct variables. Definition The el...

uh wall of text sorry about that

N3buchadnezzar

@Jordan You can also factor this polynomial by clever factorization!

NO Formulas, you just have to use your brain!

Jordan

9:12 PM

this sounds like some really advanced math to understand

Charlie

@Nimza Nice to see you too!

Jordan

I can work things out with enough time, I am just bad at memorizing algorithms

Peter Sheldrick

@Jordan, nah it just works, try it out for small examples

Nimza

@Charlie :-)

N3buchadnezzar

$$ \begin{array}{ll} f(x) & = x^3 - 3x^2 - x + 3 \\ & = x^2(x-3) - (x-3) \\ & = (x-3)(x^2-1) \\ & = (x-3)(x+1)(x-1) \end{array} $$

Jordan

9:13 PM

I cant eve nread that wikipedia page, all mathematics wikipedia pages are exclusively written for math majors

N3buchadnezzar

I always does my monic polynomials this way..

Peter Sheldrick

@Jordan, xD

N3buchadnezzar

@Jordan I find the wikipedia articles lacking in detail, and I am undergrad..

Charlie

@Nimza :D

Peter Sheldrick

@Jordan, probably :'by math majors (or undergradutes)...'

yeah some are pretty nuts

Jordan

9:15 PM

@N3buchadnezzar you say undergrad but you probably have like 4 years of college math done already

N3buchadnezzar

@Jordan Just started my second year

Jordan

what math are you up to? you probably started at calc 2 or something :P

Old John

I created some maths pages on wikipedia some years ago ... not sure if they are still there

Jordan

I don't understand the magic behind that method for factoring

N3buchadnezzar

@Jordan Complex analysis, lagrange operators, wave physics, introductory functional analysis, learning about various finite spaces. Banach, Hilbert, R^n..

Sanchez

9:17 PM

What magic?

N3buchadnezzar

@Jordan What line is the problem?

Charlie

@Nimza wassup?

Jordan

@N3buchadnezzar that is definitely not high school math :P most people start college before college algebra so the first two years of math are college algebra,trig, calc 1, calc 2

N3buchadnezzar

@Jordan University...

Jordan

I guess I get it, I just have to memorize

Nimza

9:18 PM

@Charlie I have no time for anything :( what about you?

Jordan

if I have one terms times another terms plus another terms times a term I can takes those terms and just muliply them

N3buchadnezzar

@Jordan Able to do this one? $x^3 + x^2 + 8x + 8$

Jordan

yes but I am not sure why I can make it to be $(x^2 + 8) * (x+1)$

Charlie

@Nimza i'm good. doing some stuff.

N3buchadnezzar

@Jordan It is correct :D Awesome, fantastic. What did you not completely grasp?

Jordan

9:21 PM

how to factor out an x^2 + 8

N3buchadnezzar

$x^3 + x^2 + 8x + 8 = x^2(x+1) + 8(x+1)$

Jordan

yes not how do you extract those terms without breaking rules?

N3buchadnezzar

I take it given, that you was able to do this step. Now the "clue" here is that it is easier to see the algebra if one says Oh what if $a = x+1$

Peter Sheldrick

do long division (x^3 + x^2 + 8x + 8)/(x^2 + 8)

N3buchadnezzar

@Jordan $$x^2 \cdot a + (x+1) \cdot a$$

From here you could factor out a common term, just like before. It is nothing new. It is exactly the same as when you factored out the $8$.

$a + ab = a(1+b)$

Jordan

9:24 PM

oh

it is more like I am factoring out the "a" and not the other terms

N3buchadnezzar

INDEED =D

Jordan

I should probably learn long division at some point again

N3buchadnezzar

I am looking forward to learning Galois Theory

Peter Sheldrick

definitely

do lots of examples until it sticks

Jordan

crap

now the next one is a fourth degree

$x^4 + 27x$ what is the trick?

N3buchadnezzar

9:27 PM

Stop calling it tricks!

A trick is someone pouring water over you at Halloween, if he is able to pull it of consistently it is called a technique.

Hmm, I cant factor this polynomial.. Gosh darn it, let me invent this new branch of mathematics in my diary! Hmm, I must have a gun fight tomorrow, over some lady. Oh well.

Old John

@N3buchadnezzar I found Galois theory a really satisfying topic - made all the stuff I had learned about groups and fields come together in a coherent subject

Charlie

@N3buchadnezzar why?

N3buchadnezzar

Galois actually died 28 years old, after a gun fight. Most ofhis mathematical work is contained in his diary, which he wrote on the night before he died.

@Charlie Because it is not a trick. It is a technique, which can be used in various cases and application. A trick can be used once, in a very narrow and specific case, not in a general mattter.

Charlie

hmm

N3buchadnezzar

@OldJohn Yeah, looking forward to it. Although I have so much I want to learn, and so little time.

Jordan

9:30 PM

It is a trick to me because I have never seen a problem like this, having taken calc 1 and 2 3 times

I am pretty experienced in math, four years of math and 8 math credits earned

Old John

@N3buchadnezzar same here!! - even at my age I am still learning (and have even less time!)

N3buchadnezzar

@Jordan Points at Old John's profile

Charlie

@argon

@N3buchadnezzar it's so funny when you do that

Peter Sheldrick

@OldJohn, didn't you mention somewhere you are mid-twenties or so?

N3buchadnezzar

@Charlie makes arm excusing arm movements, and move my arms around like an italian

Old John

9:32 PM

@PeterSheldrick definitely not - I can hardly remember my mid-twenties ...

N3buchadnezzar

@OldJohn My problem is that I have to take Pedagogy, and Physics, along with my math.

Charlie

@N3buchadnezzar brilliant!

Old John

@N3buchadnezzar ugh - pedagogy :(( physics is not that bad, I suppose ...

N3buchadnezzar

►

@Charlie a

Charlie

@N3buchadnezzar I love this one!!!!!!

Argon

9:34 PM

@Charlie Yes?

Jordan

this is a dirty trick and I doubt I will ever have to know how to do this

Argon

(I was making my bed)

Jordan

$x^4 + 27x$

Charlie

@Argon hmm nice

N3buchadnezzar

@OldJohn Possibly need to cut down on the adult beverages and all the female companions then ;)

user19161

9:34 PM

@Argon I was making tea.

Jordan

why would you ever want to factor that problem? it is so simple

N3buchadnezzar

@Jordan See a common term you can factor out ?

Argon

@WillHunting Yums

Old John

@N3buchadnezzar so what fun would I have left then?!?!?!

Charlie

@Argon i just like to disturb people :P

Jordan

9:34 PM

x(x^3 + 27) doesnt help me

Argon

@Charlie Really?

N3buchadnezzar

@OldJohn Exellent point

@WillHunting

@Argon yes

@Charlie

9:35 PM

@will i want tea

user19161

@OldJohn You only need 4k more to 10k, go for it!

Charlie

@argon

N3buchadnezzar

@OldJohn I did not have one slow weekend the whole semester, before the exam period began. Out friday, saturday, and most sundays.

user19161

@Charlie I want X, Y and Z.

Peter Sheldrick

and then only 10k more to 20k!

Argon

9:35 PM

@Charlie Hmmmm......

Charlie

@Argon hmhhmhmhmhhhhmmmm

Argon

@Charlie Hmfffffff

N3buchadnezzar

What subjects did you enjoy the most / do you enjoy most, a Old John ?

Charlie

@Argon pfffff

N3buchadnezzar

Points at the Hmming birds

user19161

9:36 PM

I answered several low hanging fruit recently that got me many points.

takes pictures

@Charlie geeezzzz

@Argon yeahh

@WillHunting The queeeessstiiiiiioooooooooon is how looooow caaaaaan youoooooooooo gooooooooo?

user19161

@N3buchadnezzar To the ground.

Old John

9:37 PM

hmm - different subjects at different stages ...
as an undergrad I most enjoyed number theory, complex analysis and Galois theory
later I really enjoyed potential theory
now I mostly enjoy algebraic number theory

user19161

With some luck I might make it to 10k this year, but probably not.

Old John

@WillHunting I must look out for some lhf

Charlie

@Argon wassup Aaron?

Argon

@WillHunting Maybe you could even make it to Isengard.

@Charlie Meh, nothing. I am quite bored :)

user19161

@OldJohn Yes, and you must be ready to type very fast.

Charlie

9:38 PM

@Argon me too

Jordan

oh

user19161

@Argon I wonder if I can make it to XXX. =)

Old John

@WillHunting currently aiming for my "generalist" badge - I think I just need a couple of answers for something like combinatorics maybe ...

Jordan

all I had to do was memorize a formula to solve my problem

N3buchadnezzar

@OldJohn Never heard about the two last ones

Charlie

9:39 PM

@Argon will you gain presents next saturday?

user19161

@OldJohn What's so nice about that badge?

N3buchadnezzar

@WillHunting Come on you can go lower than that.

Argon

@Charlie Nope :)

Charlie

@Argon :(

Argon

We don't really do that stuff.

N3buchadnezzar

9:39 PM

I did enjoy number theory, but I took it to early. My first semester at university

Old John

@WillHunting nothing - just seems like an interesting one to get

Charlie

@Argon oh...

Argon

I think that is Christmas influence!

Old John

@N3buchadnezzar potential theory is the study of harmonic and subharmonic functions - kept me busy for over 8 years :)

N3buchadnezzar

@Argon points at XXX swing You suure you dont do that stuff? ... wink wink

Charlie

9:40 PM

@Argon hahahah

Jordan

youtube is pretty terrible for math

user19161

I will change my username in less than 48 hours from now...

Argon

@N3buchadnezzar WOW.

@WillHunting The countdown has begun

Jordan

I need a video on how to factor a fourth degree two term non-perfect square polynomial

user19161

@Argon And then I will also share some secrets with some more people.

Argon

9:41 PM

@WillHunting Like me, perhaps?

Peter Sheldrick

@Jordan, guess roots and do long division

N3buchadnezzar

@OldJohn Yeah, I have had a little about harmonic functions and like. Analytic functions, domains and such. Nothing about subharmonic functions alas.

Peter Sheldrick

that handles most exam questions

Jordan

I don't know long division

Peter Sheldrick

time to learn it then...

user19161

9:42 PM

@Argon Hmm, I was thinking of XXX. Well, I think you know the summary already. The rest is in the details.

Jordan

I only use it like once a year, I always forget it

N3buchadnezzar

@Jordan I use the swing all the time

user19161

@Jordan You mean for numbers or polynomials?

Peter Sheldrick

then do practice examples to keep it fresh

Jordan

both

aint nobody got time for that

Peter Sheldrick

9:43 PM

well you have time to chat

Argon

HAHAHA!

user19161

@PeterSheldrick You are very pro-practice huh?

Peter Sheldrick

yes.

that sorts out exams

i'm not suggesting that leads to good math, not at all

user19161

I am not against practice, but it must come after understanding.

Peter Sheldrick

but it takes care of exams as a first step

Jordan

9:44 PM

I fail tests either way

not too concerned about that

Charlie

Hey @argon, have you ever watched this movie? this scene is hilarious!

user19161

I think it's not practice that helps in the exams but luck. You just happen to get the right question.

Peter Sheldrick

if you have wide practice then your luck increases

Argon

@Charlie With Adam Sandler, I saw it a few years ago :)

user19161

@Argon There were times in my life I looked like him.

Charlie

9:45 PM

@Argon Bam bitty bitty bitty bam bam

Argon

@WillHunting HAHAHAHA!

Old John

@N3buchadnezzar harmonic functions are fascinating - they are really well behaved inside a region where they are defined, but can get pretty interesting as you approach the boundary :)

Charlie

@WillHunting in what sense? phisically?

Jordan

instructors really like to ask trick questions on tests and those always mess me up. Like in my trig class we had to do some manipulation with a clock, but if you failed to realize that the hour hand moves something like 1/360th of a degree when the second hand does you got the question wrong

user19161

@Charlie Yes, I look different every day.

Charlie

9:47 PM

@WillHunting hmm

Jordan

it is no longer a trig question but an applications of mechanical clock theory question, you had to know that the hour hand moved steadily with the second hand, and not just at the turn of a minute or hour

user19161

@Charlie You and Jay like to say hmm.

Charlie

@WillHunting yes..hmmm

Argon

@WillHunting hmmm...

N3buchadnezzar

@OldJohn What is a function that both has a supremum and infinum on some closed interval I called?

Charlie

9:48 PM

@Argon Aaron

user19161

@Argon Aaron and I like to say XXX. =)

N3buchadnezzar

Is it common to call it a bounded function on the interval I or ?

Argon

@Charlie Marilia...!

@WillHunting XXX

Old John

@N3buchadnezzar I would call it that, yes

user19161

@N3buchadnezzar We just say it is bounded.

N3buchadnezzar

9:49 PM

►

Charlie

@Argon Oh, i forgot to tell you that my name comes , somehow, don't ask me how, from "Myriam"

@N3buchadnezzar I looooove this!

Argon

@Charlie Oh, I guess that makes sense then!

Jordan

well I have already met my limits of understanding of division, why can't I get a factor out of this? $x(x^3 + 27)$

Argon

Huh.

Charlie

@Argon why?

Argon

9:51 PM

@Charlie I was wondering what it could have came from

Charlie

@N3buchadnezzar "When I'm alone I F* myself!"

user19161

@Jordan Well you need to knw that $a^3+b^3=(a+b)(a^2-ab+b^2)$.

Argon

Difference of cubes

user19161

@Jordan Similarly, $a^3-b^3=(a-b)(a^2+ab+b^2)$.

N3buchadnezzar

@OldJohn I had a misconception about that if $\lim_{x\to a}f(x)$ existed, this implied that the function was well behaved on a small enough interval I around a. Where well behaved implies continuity and boundedness. It was fun finding a counter example ^^

Peter Sheldrick

9:52 PM

@Jordan, hint: 3^3=27

Argon

@N3buchadnezzar What counterexample?

Old John

@N3buchadnezzar ah yes - limits can exist for some pretty crazy functions :)

user19161

@N3buchadnezzar Of course it is bounded in some interval around $a$, but it is not necessarily continuous at $a$.

Charlie

Once i was spending a good time trying to prove something.... it happens that it was a false statement.... i realized that when i found a counter example

N3buchadnezzar

@Argon Try to find one ;)

Jordan

9:54 PM

@WillHunting Is that somethingto just memorize?

N3buchadnezzar

@WillHunting Does it really have to be bounded? Like John said, there exists some pretty crazy functions.

user19161

@Argon Well just take $f(0)=1$ and $f(x)=0$ otherwise, totally trivial.

user19161

@N3buchadnezzar Like I said, it has to be bounded in some interval around $a$ if the limit exists, this is trivial.

Argon