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Does there exist a basis $(p_0,p_1,p_2,p_3)\in P_3(\Bbb F)$ such that none of the polynomials $p_0,p_1,p_2,p_3$ has degree $2$?

First thing is trying to understand what is being asked of me.

Is there a list of vectors that are polynomials that are linearly independent and spanning of $P_3(\Bbb F)$ such that we don't have some $az^2$?

Well to span $P_3(\Bbb F)=a_0+a_1 z + a_2 z^2 + a_3 z^3$ of dimension $4$, we need four vectors $\{(1),(z),(z^2),(z^3)\}$, and hence we must have a $p_i$ for some $i=\{0,1,2,3\}$ in the basis as a degree $2$ polynomial.

Is this a valid proof?

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    $\begingroup$ $\{z^2+z^3, z^3\}$ has $z^2$ in its span. $\endgroup$
    – David Mitra
    Commented Jan 17, 2015 at 13:01
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    $\begingroup$ @DavidMitra Ahhh You have me, thank you. 'Accepts davids comment as the best answer' $\endgroup$
    – Wow you have a good name
    Commented Jan 17, 2015 at 13:07

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