We can find our polynomial interpolation by writing g as a general polynomial of the given degree,expressing each of our conditions as a relation among  g's coefficients, and solving the resulting system of linear equations to determine these coefficients. This can be done similarly for eitherproblem; the only difference is that in one case some of the equations come from derivatives of g.

We start this process for our first problem where the derivatives of g must be continuous at b and c.

Set

g(x) = g₀ +g₁ x+g₂ x²+g₃ x³.

Our four conditions become:

g₀+ g₁ b + g₂ b² + g₃ b³ = f(b)

g₁ + 2g₂ b + 3g₃ b² = f '(b)

g₁ + 2g₂ c + 3g₃ b² = f '(c)

g₀ + g₁ c + g₂c² + g₃ c³ = f(c)

In the second problem there are k equations, each  like the first and fourth here going up to degree k - 1.

g₀ + g₁ x_j + g₂ x_j² +...+ g _{k -1} x_j^k-1 = f(x_j ) 

The pedestrian approach is to write down all these equations and use algebraic manipulations to solve them.