 Even and odd polynomial density on a positive interval

On a positive compact interval, both even-power and odd-power complex polynomial families are uniformly dense in the continuous-function algebra. The even family is a conjugation-closed unital algebra separating points, since squaring is injective on the interval; the Stone-Weierstrass theorem proves its density. Equivalently use the homeomorphism $t\mapsto t^2$ and polynomial approximation on $[a^2,b^2]$. For the odd family, given a target $f$, first approximate $f/t$ by an even polynomial $p(t^2)$. Then $tp(t^2)$ is odd and its error is at most $b$ times the first error. Division is continuous because the interval avoids zero. Thus the odd family can be dense even though it is not a unital subalgebra.