= Solution
\b[The even-power family is dense on $[1,2]$.] It is the algebra $A=\{p(t^2):p\in\mathbb C[s]\}$. It contains constants, is closed under <complex conjugation>, and separates points because the real function $t\mapsto t^2$ is injective on this positive interval. The complex Stone-Weierstrass theorem therefore applies.
Equivalently, the squaring map is a <homeomorphism> from $[1,2]$ onto $[1,4]$. Given $f\in C([1,2])$, approximate the <continuous function> $s\mapsto f(\sqrt s)$ uniformly on $[1,4]$ by complex <polynomials>, using the <Weierstrass approximation theorem> on the real and imaginary parts. Substitution $s=t^2$ produces exactly the required even-power approximants. The positive interval is what lets even powers distinguish all its points.
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