1. Find Lagrange polynomials that approximate f(x)=x^3 (a) Find the linear interpolation polynomial P_(1)(x) using the nodes x_(0)=-1 and x_(1)=0. (b) Find the quadratic interpolation polynomial P_(2)(x) using the nodes x_(0)=-1 x_(1)=0 and x_(2)=1 (c) Find the cubic interpolation polynomial P_(3)(x) using the nodes x_(0)=-1,x_(1)=0 x_(2)=1 and x_(3)=2 (d) Find the linear interpolation polynomial P_(1)(x) using the nodes x_(0)=1 and x_(1)=2 (e) Find the quadratic interpolation polynomial P_(2)(x) using the nodes x_(0)=0, x_(1)=1 and x_(2)=2

(a) The linear interpolation polynomial using the nodes and is given by: Substituting the values, we get: (b) The quadratic interpolation polynomial using the nodes , , and is given by: Substituting the values, we get: (c) The cubic interpolation polynomial using the nodes , , , and is given by: Substituting the values, we get: (d) The linear interpolation polynomial using the nodes and is given by:$P_{1}(x) = \frac{(x - x_{1})}{(x_{0} - x_{1})}f(x_{0}) + \frac{(x