PROBLEM 5: (10 points) Define the polynomial p(t)=1+t++tn1 with coefficients all 1's. We shall try to recover these coefficients from the n values of the polynomial p(i+1),i=1,,n. Note that p(i+1)=j=1n(1+i)j1=i1[(1+i)n1]. If the coefficients of the polynomial are denoted by x1,,xn, we write i1[(1+i)n1]=p(i+1)=j=1n(i+1)j1xjAx=b, where A=(ai,j),ai,j=(1+i)j1,bi=i1[(1+i)n1]. Write a program to solve the resulting system using GE. Run this program for n=4,5,6,7,8,9,10,,20. Create a table for the obtained coefficients x1,,xn (which we know are all 1's). Describe your observations. (Hint: The difficulties you observe are due to the fact that the matrix A is ill-conditioned, i.e. after a certain value of n the roundoff error is propagated and magnified and thus most of the computed values for xi are not correct.)