#Complex numbers
rustic zephyrBOT
jaunty depot
Z times the conjugate of z is |z|^2
Reducing it to a single variable inequality
And it should give you a cubic in |z|
Multiply both sides by the denominator
To get the cubic
Nah just substitute in |z|^2 at the start
No reason to multiply by conjugate
yeah then just reduce to a cubic
subtract (|z|^2 - 4) from both sides
u have a cubic in |z|
yeah, then solve for |z| and remember |z| >= 0
sage flumeBOT
Infinity
dapper whale
I would try to find the zero roots then
in terms of |z|
you can guess one pretty easily between -2, -1,0,1,2
Notice |z| = -2 is a double root
And notice that the factor of |z|³ is positive
Like this is what the cubic function kinda looks like
So it's actually only |z| ≤ 3
\sqrt
sage flumeBOT
Infinity
dapper whale
you forgot \
dapper whale
Because looking at the inequality this is basically a cubic function like in the sketch and the inequality is asking for all solutions that result to either 0 or negative numbers
So looking at the behavior of the function it is only negative or equal to zero at |z| <=3
but very well done of you👍🏻
dapper whale
deutsch haah
yes
at my university we can try to guess a zero root in the range of -2 to 2 (integers)
other than that factorizing cleverly is also a good but difficult skill
to get a faktorzerlegung
You can zerlegen it in faktoren geschickt like (x+2)²(x-3)
nein man echtnicht
es gibt eine technik wie man das in faktoren zerlegen kann aber ich kenn die selber nicht
natürlich
wie gesagt gut gemacht!
angewandte Mathematik
Sowas hab ich nicht wirklich aber man könnte mein Informatik haha
visual code ich sehes
welchen beweis nochmal?
ich wollte anmerken bei $c_i = 0$ dass daraus folgt dass $a_i = 0$ ist ein bisschen gefährlich, weil wenn du die potenzen aufschreibst, dann merkst du dass im nenner durch 0 geteilt wird
sage flumeBOT
adonhs
