Hi, I'm working on the following problem and was wondering if anyone could check to see if I've solved it correctly (I can't help but wonder if my method is incorrect - it seems a little too simple).

Define $\displaystyle T : P_{3}(\mathbb{R}) \rightarrow P_{3}(\mathbb{R}) $ by $\displaystyle T(f)(x) = xf^{\prime}(x) $ . Find the range of $\displaystyle T $

My attempt is as follows:

Let $\displaystyle f(x) = x^{3} + x^{2} + x + 1 $ (standard form for a polynomial of $\displaystyle P_{3}(\mathbb{R}) $)

Then $\displaystyle T(f)(x) = x(3x^{2} + 2x + 1) = 3x^{3} + 2x^{2} + x $

so range $\displaystyle T = span \{ x^{3}, x^{2}, x \} $

It seems a little crude to me for some reason...

Is it correct?

Thanks,