Find three subspaces $W_1$, $W_2$ and $W_3$ of $k [X]$ such that $k[X]\cong W_1\oplus W_2\oplus W_3$

Question

I am trying to find three non-trivial subspaces of all polynomials, there are infinite subspaces of the space of all polynomials? I could fix a natural $n$ and say that the polynomials of degree less than $n$ are a subspace, those of degree equal to n other and those of degree greater than $n$ another and in this case could not arm the direct sum with those three subspaces?

Answer 1

Hint:

$k\bigl[X^3\bigr]$ could be one of them. Can you proceed from there?

Answer 2

A basis of $k[X]$ is $1, X, X^2, X^3, \dots$. The only thing you have to do is divide the basis elements in three groups, so for instance $$\begin{align*} W_1 &= \text{span}(1) = k \\ W_2 & = \text{span}(X) = k \cdot X \\ W_3 & = \text{span}(X^2, X^3, \dots) = X^2 \cdot k[X]\\ \end{align*} $$ would work.