Let y (x)= ax3 + bx2 + cx + d
y(0) = 1
0 + 0 + 0 + d=1,
So d = 1
y(1) = 0
a + b + c + 1=0
a + b + c = -1
y(2) = 1
8a + 4b + 2c + 1=1
8a + 4b + 2c = 0
y(3) = 10
27a + 9b + 3c + 1=10
27a + 9b + 3c = 9
9a + 3b + c = 3
a + b + c = -1 ..........(1)
8a + 4b + 2c = 0 ...........(2)
9a + 3b + c = 3 ...........(3)
From eqn. 1 and eqn. 3:
a - 9a + b - 3b + c - c = -1 - 3
-8a - 2b = -4
-4a - b = -2
4a + b = 2
eqn. 2 - and eqn.3:
8a - 18a + 4b - 6b + 2c - 2c = 0 - 6
-10a - 2b = -6
-5a - b = -3
5a + b = 3
subtract 5a + b = 3 from 4a + b = 2:
4a - 5a + b - b = 2 - 3
-a = -1
a = 1
substitute a = 1 into 4a + b = 2:
4 + b = 2 --> b = -2
subsitute a = 1 and b = -2 into a + b + c = -1:
1 - 2 + c = -1
-1 + c = -1
c = 0
So cubic polynomial equation is:
y (x)= x3 - 2x2 + 1