Linear transform of polynomials

The following question is from Golan's linear algebra book. I have posted a solution in the answers.

Problem: Let $F$ ba field and let $V$ be a vector subspace of $F[x]$ consisting of all polynomials of degree at most 2. Let $\alpha:V\rightarrow F[x]$ be a linear transformation satisfying

$\alpha(1)=x$

$\alpha(x+1)=x^5+x^3$

$\alpha(x^2+x+1)=x^4-x^2+1$.

Determine $\alpha(x^2-x)$.

Answer

Your calculation looks fine. Equivalently, but with rather less work, note that
$$x^2-x=(x^2+x+1)-2(x+1)+1,$$
so we can compute $\alpha(x^2-x)$ directly by linearity, without computing $\alpha$ at the most common basis elements.