Question

Create a table to find the third differences for the polynomial s^3 − s^2 +s for integer values of ss from −3 to 3.

Answer 1

Explanation:

`f(s)=s^3-s^2`

at s = -3

`f(-3)=-3^3-(-3)^2=18`

at s = -2

`f(-2)=-2^3-(-2)^2\\\Rightarrow f(-2)=-12`

at s = -1

`f(-1)=-1^3-(-1)^2\\\Rightarrow f(-1)=-2`

at s = 0

`f(0)=0^3-0^2\\\Rightarrow f(0)=0`

at s = 1

`f(1)=1^3-1^2\\\Rightarrow f(1)=0`

at s = 2

`f(2)=2^3-2^2\\\Rightarrow f(2)=4`

at s = 3

`f(3)=3^3-3^2\\\Rightarrow f(3)=18`

First difference

`f(-2')=f(-2)-f(-3)=-12-18=-30`

`f(-1')=f(-1)-f(-2')=-2--12=10`

`f(0')=f(0)-f(-1)=0--2=2`

`f(1')=f(1)-f(0)=0-0=0`

`f(2')=f(2)-f(1)=4-0=4`

`f(3')=f(3)-f(2)=18-4=14`

Second difference

`f(-1'')=f(-1')-f(-2')=10--30=40`

`f(0'')=f(0')-f(-1')=-8-40=-48`

`f(1'')=f(1')-f(0')=-2--8=6`

`f(2'')=f(2')-f(1')=14-4=10`

Third difference

`f(0'')-f(-1'')=-8-40=48`

`f(1'')-f(0'')=-2--8=6`

`f(2'')-f(1'')=4--2=6`

`f(2'')-f(1'')=10-4=6`

18 -12 -2 0 0 4 18

-30 10 2 0 4 14

40 -8 -2 4 10

-48 6 6 6