In this problem, we walk you through an alternative to Lagrange interpolation. (a) Let's say we wanted to interpolate a polynomial through a single point, (x0, yo). What would be the polynomial that we would get? (This is not a trick question.) (b) Call the polynomial from the previous part fo(x). Now say we wanted to define the polynomial fi(x) that passes through the points (x0,yo) and (x1,y). If we write fi(x) = fo(x)+a1(x xo), what value of aj causes f1(x) to pass through the desired points? (c) Now say we want a polynomial f2(x) that passes through (x0,yo), (x1,y), and (x2,y2). If we write f2(x) = fi(x) +a2(x xo)(x - x1), what value of a2 gives us the desired polynomial? (d) Suppose we have a polynomial fi(x) that passes through the points (x0, yo), ..., (xi,yi) and we want to find a polynomial fi+1(x) that passes through all those points and also (Xi+1,Yi+1). If we define fi+1(x) = fi(x) +di+1 T1;=0(x xj), what value must dit1 take on