Studying for a final and came accross this problem in the textbook. Considering I have no idea how to even start im a bit scared :). Any explanation would be greatly appreciated.
problem: If f(x)is a polynomial of degree m, show that f(x) may be expressed in the form $$f(x)=\sum_{r=0}^m c_rL_r(x)$$ with $c_r=\int_{0}^\infty e^{-x}L_r(x)f(x)dx$.
Deduce that $\int_{0}^\infty e^{-x}x^kL_n(t)dt=0$ if $k<n$
and $\int_{0}^\infty e^{-x}x^kL_n(t)dt=0$ if $k=n$