# The vector space of polynomials

I was given a theorem:

The polynomials (where $f$ and $g$ are complex polynomials of degrees $n$ and $m$)

$$f(z), zf(z), \ldots , z^{m−1}f(z), g(z), zg(z), \ldots,z^{n−1}g(z)\tag{7.6.4}$$

form a basis of the space $P(\mathbb{C},m + n − 1)$ if and only if $f$ and $g$ have no common zero.

And also a problem:

Find a basis, and the dimension, of the space of polynomials spanned by the polynomials in $(7.6.4)$ when $f(z) = z^2(z − 1)$ and $g(z) = z^2(z + 1)$

Now, obviously, $f$ and $g$ have a common zero, namely - $0$. So I cannot use the theorem, can I? If no, what the other way around it?

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What is $P(\mathbb{C},m+n-1)$? – Dedalus Jun 16 '13 at 18:59
Yeah, sorry! This is the space of complex polynomials of degree at most $n+m-1$ – Sarunas Jun 16 '13 at 19:01
In that case - you can't use the theorem above no, they have a common zero. But you don't have to use this to compute the dimension or find a basis of the space spanned by the polynomials above! – Dedalus Jun 16 '13 at 19:05
I have a question: is the dimension for this space 2? As if indeed it is, the problem is easier than I think. if not, I'm kind of lost :) – Sarunas Jun 16 '13 at 19:10
You have $m = n = 3$. Put that and $f$ and $g$ in $(7.6.4)$. The point of the problem is probably to show you that some of the obtained set of polynomials is not independent. Eliminate those and you have your basis (and a dimension). – Vedran Šego Jun 16 '13 at 21:42