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) To find the linear interpolation polynomial using the nodes and , we can use the formula for the Lagrange polynomial: Substituting the given values, we have: (b) To find the quadratic interpolation polynomial using the nodes , , and , we can use the formula for the Lagrange polynomial: Substituting the given values, we have: (c) To find the cubic interpolation polynomial using the nodes , , , and , we can use the formula for the Lagrange polynomial: Substituting the given values, we have: (d) To find the linear interpolation polynomial using the nodes and , we can use the formula for the Lagrange polynomial:$P_{1}(x) = \frac{(x-x_{1})}{(x_{