Saturday, 6 September 2014

Transformation of Roots for a Cubic Polynomial

Consider x3+px+q=0
with roots α,β.γ




We want to find the degree 3 polynomial that has roots :
αβγ,αγβ,βγα



My attempt so far:



αβγ=αβγγ2=qγ2
and similarly for all of the others:



So now I let t=qx2 and substitute into the original polynomial; except I will multiply everything by x first to make my substitution easier.




x4+px2=qx
q2t2pqt=qqt
Squaring both sides and multiplying by t4, we obtain:
qt3+p2t22pqt+q2=0



Except this seems like a messy, tedious solution as you have to muck around with square roots etc. Is there a nicer way to do this?

No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...