Here's 3
5 4 - 1 -10 -23 -40
- 1 -5 - 9 -13 -17
-4 -4 -4 -4
Since the second differences are all the same, this is a quadratic
To find the polynomial, we have this system :
a + b + c = 5
4a + 2b + c = 4
9a + 3b + c = -1
Subtract the first equation from the second and the second equation from the third and we have
3a + b = -1
5a + b = -5 subtract the first equation from the second and we have
2a = -4
a = -2
And using 3a + b = -1 to find b, we have.......3(-2) + b = -1 → b = 5
And using a + b + c = 5 to find c we have -2 + 5 + c = 5 → c = 2
So the generating polynomial is : -2x^2 + 5x + 2
Here's 4
20 4 0 20 76 180
-16 -4 20 56 104
12 24 36 48
12 12 12
Since the third differences are the same, this is a cubic polynomial
And to find the generating polynomal, we have this system
a + b + c + d = 20
8a + 4b + 2c + d = 4
27a + 9b + 3c + d = 0
64a + 16b + 4c + d = 20
Solving this system in a similar manner to (3) we have that a = 2, b = -6, c = -12, d = 36
So the generating polynomial is 2x^3 - 6x^2 -12x + 36