Question

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.......

Answer 1

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.

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

Do you mean real and imaginary part?

Answer 8

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

Answer 9

and not mess up the reals lol

Answer 10

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

Answer 11

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 :)

Answer 12

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

Answer 13

wha!? arithmetics gone fundamental too?

Answer 14

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.
\[x^2+4 = (x+4i)(x-4i)\]

Answer 15

Oh amistre isnt that partial fraction decomposition?

Answer 16

yes it is; yes it is ;)

Answer 17

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

Answer 18

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