Find a polynomial of minimal degree which has the following

Find a polynomial of minimal degree which has the following graph

Solution

Threaph touches x axis : x = -1 ( multiplicity 2)

Crosses the x axis : x = 1 and x = 2 ( multiplicity 1)

Let f(x) = a(x+1)^2(x -1)(x -2)

Plug x =0 ; f(0) = -1

-1 = a*1*(-1)*(-2)

a = -1/2

f(x) = (-1/2)(x+1)^2(x -1)(x -2)