amistre64
  • amistre64
i learnt the other day that higher degree functions can always be reduced to linear and irreducible quadratic parts... but how to state that into a question, hmmm.......
Mathematics
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jamiebookeater
  • jamiebookeater
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nowhereman
  • nowhereman
The fundamental theorem of algebra states that in complex numbers any polynomial can be written as a product of linear parts. So because for polynomials with real coefficients alongside each root it's complex conjugate is a root too, those two linear parts \[x-x_0\;\text{and}\; x-\overline{x_0}\] will yield a quadratic part.
amistre64
  • amistre64
yeah, thats what my teacher told me lol; I was like: thers a fundamental theorum of algebra? lol
nowhereman
  • nowhereman
and the funny thing is, that mostly it is proven using analytic methods

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amistre64
  • amistre64
can you decompose fraction into complex parts without messing things up in the end?
nowhereman
  • nowhereman
I don't know what kind of a decomposition you are talking about, can you give an example?
amistre64
  • amistre64
\[\frac{10x + 7}{(x-1)(x^2+4)}\] for example..
nowhereman
  • nowhereman
Do you mean real and imaginary part?
amistre64
  • amistre64
imaginary parts; can this be decomposed into imaginary parts and not mess with the real solutions?
amistre64
  • amistre64
and not mess up the reals lol
amistre64
  • amistre64
the quad is irreducible unless you use imaginaries; and I was curious if we could break it down to all linear stuff
amistre64
  • amistre64
i mean I guess I could take the time to actually do it, but then id have to get a pencil out and find some paper and ....its just alot of work :)
anonymous
  • anonymous
I was surprised to find earlier this year that there was a fundamental theorem of arithmetic.
amistre64
  • amistre64
wha!? arithmetics gone fundamental too?
nowhereman
  • nowhereman
Oh yeah, that is correct. And because the real numbers are embedded into the complex, that won't change anything about the solutions. In fact the solution formula for higher dimensional polynomials are mostly written with complex numbers in mind. So you would get. \[x^2+4 = (x+4i)(x-4i)\]
anonymous
  • anonymous
Oh amistre isnt that partial fraction decomposition?
amistre64
  • amistre64
yes it is; yes it is ;)
nowhereman
  • nowhereman
ähm, sorry should be 2s instead of the 4s there ;-)
amistre64
  • amistre64
they are imaginary 4s lol..so it doesnt matter ;)

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