Pal Approximation Theorem

We prove the Pal polynomial approximation theorem which asserts that given a continuous function on a symmetric interval we can approximate it uniformly by a polynomial of the form f(0)+ a_1 x + a_2 x^2 + ... + a_n x^n + (higher terms) for any coefficients a_1, ..., a_n that we wish! This theorem uses the Weierstrass Approximation Theorem and linear interpolation to achieve this beautiful theorem.

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[olivierbegassat]

I don't understand the initial set up of the theorem. It seems equivalent to stating that any continuous function on [-1, +1] which vanishes at the origin can be uniformly approximated by a polynomial of the form x^(n+1) * Q(x). What's the purpose of the a_1, a_2, a_n ? a_0 is there to kill the value at 0 but as soon as you fix the other coefficients you have essentially just modified the underlying f to become some alternative function g that is continuous and vanishes at 0.

Also in your proof I am confused that you would use the name q for what seems to be a different q than from the theorem statement.