**Exercise**: do the polynomials \(p_0(x)=1,~~p_1(x)=(x-1),~~p_2(x)=(x-1)^2,~~p_3=(x-1)^3\) form a basis of the vector space of cubic polynomials (with real coefficients)? If so, express \(x^3\) in this basis.

**Solution**: use the binomial: \(x^n = \big((x-1)+1\big)^n = \sum_{m=0}^n {n\choose m}(x-1)^m\) to infer …