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amistre64

  • 5 years ago

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.......

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  1. nowhereman
    • 5 years ago
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    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.

  2. amistre64
    • 5 years ago
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    yeah, thats what my teacher told me lol; I was like: thers a fundamental theorum of algebra? lol

  3. nowhereman
    • 5 years ago
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    and the funny thing is, that mostly it is proven using analytic methods

  4. amistre64
    • 5 years ago
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    can you decompose fraction into complex parts without messing things up in the end?

  5. nowhereman
    • 5 years ago
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    I don't know what kind of a decomposition you are talking about, can you give an example?

  6. amistre64
    • 5 years ago
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    \[\frac{10x + 7}{(x-1)(x^2+4)}\] for example..

  7. nowhereman
    • 5 years ago
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    Do you mean real and imaginary part?

  8. amistre64
    • 5 years ago
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    imaginary parts; can this be decomposed into imaginary parts and not mess with the real solutions?

  9. amistre64
    • 5 years ago
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    and not mess up the reals lol

  10. amistre64
    • 5 years ago
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    the quad is irreducible unless you use imaginaries; and I was curious if we could break it down to all linear stuff

  11. amistre64
    • 5 years ago
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    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 :)

  12. anonymous
    • 5 years ago
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    I was surprised to find earlier this year that there was a fundamental theorem of arithmetic.

  13. amistre64
    • 5 years ago
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    wha!? arithmetics gone fundamental too?

  14. nowhereman
    • 5 years ago
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    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. \[x^2+4 = (x+4i)(x-4i)\]

  15. anonymous
    • 5 years ago
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    Oh amistre isnt that partial fraction decomposition?

  16. amistre64
    • 5 years ago
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    yes it is; yes it is ;)

  17. nowhereman
    • 5 years ago
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    ähm, sorry should be 2s instead of the 4s there ;-)

  18. amistre64
    • 5 years ago
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    they are imaginary 4s lol..so it doesnt matter ;)

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