You mean writing ##ax^3 + bx^2 + cx + d = p## which will give us system of four equations with the same LHS and RHS will always be the result for the individual elements of the basis?
So the first equation would be ##ax^3 + bx^2 + cx + d = c##, second ##ax^3 + bx^2 + cx + d = d##, third ##ax^3...