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Vector space, direct sums involving linear maps and polynomials

Let be $$V$$ a vector space over $$K$$ and lets $$a\in K\setminus\{0\}$$ and $$T:V \to V$$ a linear map s. t. $$a^2T-3aT^2+T^3=0$$. Show that $$V = Ker(T) \oplus Im(T)$$.

My professor told me an observation:

This result is more general, that is, if I have a polynomial $$P$$ such with $$P(0) = 0$$ and $$P'(0) \neq 0$$ such that $$P(T) = 0$$ then $$V = Ker(T) \oplus Im(T)$$.

However, I didn’t find the relation with the exercise and his observation (of course, that polynomial assets the assumption but???). Can you give me a tip to solve this exercise??

Also I’ve tried to show that $$T$$ is a projection, but it didn’t work.