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In \(\mathbb{R}_3[X]\), recall the decomposition into even and odd polynomials: \(\mathbb{R}_3[X] = P \oplus I\) where \(P = \mathrm{Vect}(1, X^2)\) (even) and \(I = \mathrm{Vect}(X, X^3)\) (odd). Verify the direct sum and write a basis of \(\mathbb{R}_3[X]\) adapted to this decomposition.
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