The generic equation of a third degree polynomial is given by: y = ax ^ 3 + bx ^ 2 + cx + d We must make a system of equations to find the values of a, b, c, d. We have then: For (1, 3): 3 = a (1) ^ 3 + b (1) ^ 2 + c (1) + d 3 = a + b + c + d For (2, -2): -2 = a (2) ^ 3 + b (2) ^ 2 + c (2) + d -2 = 8a + 4b + 2c + d For (3, -5): -5 = a (3) ^ 3 + b (3) ^ 2 + c (3) + d -5 = 27a + 9b + 3c + d For (4.0): 0 = a (4) ^ 3 + b (4) ^ 2 + c (4) + d 0 = 64a + 16b + 4c + d We obtain the following system of equations: 3 = a + b + c + d -2 = 8a + 4b + 2c + d -5 = 27a + 9b + 3c + d 0 = 64a + 16b + 4c + d Whose solution is: a = 1 b = -5 c = 3 d = 4 The polynomial will then be: y = x ^ 3 - 5x ^ 2 + 3x + 4 Answer: y = x ^ 3 - 5x ^ 2 + 3x + 4
