Find the eigenvalues of this endomorphism of R[X]

[penguin007]

Homework Statement

f is an endomorphism of Rn[X]
f(P)(X)=((aX+b)P)'

eigenvalues of f?

Homework Equations

(a,b)<>(0,0)

The Attempt at a Solution

If a=0, then f(P)=bP', and only P=constant is solution

if a<>0, then I put Q=(ax+b)P, f(P)=cP is equivalent to (ax+b)Q'=Q (E)

I solved (E) and found Q(X)=(aX+b)^c but then if I say P(X)=(aX+b)^(c-1), I can't find c...

 

[vela]

Is f(P)=(aX+b)P' or [(aX+b)P]'? You wrote it both ways.

 

[penguin007]

f(P)=[(aX+b)P]'

 

[vela]

Sorry, I misread your initial post. I see what you did now. Your solution for Q(X) should be $Q(X)=(aX+b)^{c/a}$ so $P(X)=(aX+b)^{c/a-1}$. Do you see now what values c can be?

 

[penguin007]

Thanks a lot vela, I mistook when I solved (E)... Now I can see the values for c (s.t c/a-1 is integer and therefore P is a polynom)...