factor x^4 + 1. over non reals

Question

myininaya · Answer

$$(x^2-1)(x^2+1)=(x-1)(x+1)(x-i)(x+i)$$

perl · Answer

thats wrong

perl · Answer

x^4 +1 = (x^2)^2 - i ^2 = (x^2 + i)(x^2-i)

myininaya · Answer

oops i thought that was a minus sign in between x^4 and 1

myininaya · Answer

$$(x^2-i)(x^2+i)$$

perl · Answer

so im having trouble factoring x^2 + i

myininaya · Answer

my bad!
its early for me and i don't have my glasses on lol

myininaya · Answer

how do you know when to stop factoring?

anonymous · Answer

$$(x^2 + 1)^2 = (x^2+i)(x^2-i) $$

perl · Answer

factor it into a+bi , roots

perl · Answer

youre not there yet , lol

perl · Answer

this is a delight

anonymous · Answer

$$x = \pm e^{i \pi/4}$$

perl · Answer

should be 4 roots

perl · Answer

,how did you get that evern

anonymous · Answer

sorry ... x = \pm e^{i (\pi/4 +\pi/2)}

anonymous · Answer

use euler's form e^ix..
this allows us to calculates roots easily( a root is ^1/k)

anonymous · Answer

smoething wrong with the renderer ... $$x = \pm e^{i (\pi/4 +\pi/2)}$$

anonymous · Answer

that was the 2nd set of sols

anonymous · Answer

that was the 2nd set of sols

anonymous · Answer

was i any help?

perl · Answer

yes very much , one sec

perl · Answer

how did you know it was pi/4

perl · Answer

we have x = sqrt (i) , -sqrt (i) , sqrt (-i) , -sqrt(-i)

anonymous · Answer

imagine the complex plane the imaginary axis,which denotes $$ \pi/4 $$ is at an angle of 90 degrees to the real axis => 90/2 = 45 =>pi/4 $$i = e^{i \pi/2}$$

perl · Answer

i dont follow

perl · Answer

the imaginary axis is pi/2 to the real axis

anonymous · Answer

yes...and then we sq root it.The square root expressed as an exponential is x^(1/2) hence the entire power bit of the exponent is divided by 2 giving pi/4

perl · Answer

ok so you did x^2 = e^i(pi/2 +2pik )

perl · Answer

, for x^2 = i

anonymous · Answer

yes...and then we sq root it.The square root expressed as an exponential is x^(1/2) hence the entire power bit of the exponent is divided by 2 giving pi/4

perl · Answer

there are actually an infintie number of solutions, but we are interested in the principal

anonymous · Answer

yes...i actually skipped out on generalizing 2pi phase factor

perl · Answer

ok and this follows from the fact that a + b i = r e^i(theta + 2pik) ?

perl · Answer

, which is euler identity

perl · Answer

ok so then its just a matter of exponentiation , x = e^[i(pi/2 + 2pik)*1/2]

anonymous · Answer

yes....again while doing the initial pass over the question (complex analysis)  i usually omit the 2pi factor.It necessary for the later steps but it just makes a mess while trying to understand the solution.

anonymous · Answer

you got it

perl · Answer

we could have also used the trig form of a + bi = rcos t + i sin t , but this is cleaner

anonymous · Answer

:)

perl · Answer

then wed have to use demoivres theorem, exponential is faster

anonymous · Answer

once again an astute observation with a firm logical reason backing it up

perl · Answer

yes i dont even remember demoivrs,

perl · Answer

this is more concise ;)

perl · Answer

so i have x = +- e^[i(3pi/4)], x = + - e^[i(pi/4)]

perl · Answer

, why isnt it acceptable to just write the solutions as above

anonymous · Answer

what grade are you in?

perl · Answer

x = sqrt (i) , -sqrt (i) , sqrt (-i) , -sqrt(-i)

perl · Answer

im in college ;)

perl · Answer

my book gives radical solutions , sqrt2/2 + i sqrt 2 / 2 , etc

anonymous · Answer

ahh...then there should be no problem in expressing the answer as an exponent

perl · Answer

no i mean, the book didnt leave it in exponential euler form

anonymous · Answer

books generally use a computer program to solve their questions and programs rarely try an make stuff easier for people to understand (I should know as i'm doing engineering in computer science)

anonymous · Answer

books generally use a computer program to solve their questions and programs rarely try an make stuff easier for people to understand (I should know as i'm doing engineering in computer science)

perl · Answer

so basically they want 
sqrt i = a + b i , for some a,b , and they got sqrt2/2 + sqrt2/2 i , or something

perl · Answer

apparently its not good to leave it as i^(1/2) as a solution

anonymous · Answer

if you insist....here is where we expnad eulers form $$e^{ix} = cos(x) + i sin(x)$$

perl · Answer

there we go

perl · Answer

so sometimes we use all three forms

anonymous · Answer

cos(3pi/4) = -1/sqrt(2) ,sin(3pi/4) = 1/sqrt(2) .....i'm not sure about the exact values just check them on a calculator

perl · Answer

, err, two forms , etc

anonymous · Answer

yes sometimes depending on what you need easy additive manipulation or easy multiplicative manipulation you use different forms

perl · Answer

thx are you good with 3d graphers , i have this website called 
romanlabs.com , i found a 3d grapher but i cant figure out the code input

perl · Answer

i also have a calculus problem, 
Find the derivative of
f (x, y) = xy/ (x+ y)
at the point (2, -1) in the direction of 6i + 8j . i got it started 
the partials are

perl · Answer

< y^2/(x+y)^2, x^2/(x+y^2) >

perl · Answer

it would be sweet if i could figure out how to use this graphing calculuator

anonymous · Answer

there are ways of graphing complex numbers (we usually express the imaginary diemensions as colors)

anonymous · Answer

p.s. if you want to see the function use wolfram alpha (online .....realy good) or gnuplot(offline allows more flexibility but is more complex) ...note both service/programs are free/open source.

anonymous · Answer

anyway must leave if you have any other related questions feel free to fb me (note: my fb profile has my name misspelled so look for "Erevn")