## 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.......

1. 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.

2. amistre64

yeah, thats what my teacher told me lol; I was like: thers a fundamental theorum of algebra? lol

3. nowhereman

and the funny thing is, that mostly it is proven using analytic methods

4. amistre64

can you decompose fraction into complex parts without messing things up in the end?

5. nowhereman

I don't know what kind of a decomposition you are talking about, can you give an example?

6. amistre64

$\frac{10x + 7}{(x-1)(x^2+4)}$ for example..

7. nowhereman

Do you mean real and imaginary part?

8. amistre64

imaginary parts; can this be decomposed into imaginary parts and not mess with the real solutions?

9. amistre64

and not mess up the reals lol

10. amistre64

the quad is irreducible unless you use imaginaries; and I was curious if we could break it down to all linear stuff

11. 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 :)

12. anonymous

I was surprised to find earlier this year that there was a fundamental theorem of arithmetic.

13. amistre64

wha!? arithmetics gone fundamental too?

14. 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. $x^2+4 = (x+4i)(x-4i)$

15. anonymous

Oh amistre isnt that partial fraction decomposition?

16. amistre64

yes it is; yes it is ;)

17. nowhereman

ähm, sorry should be 2s instead of the 4s there ;-)

18. amistre64

they are imaginary 4s lol..so it doesnt matter ;)