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