Step-by-step explanation:
[tex]f(s)=s^3-s^2[/tex]
at s = -3
[tex]f(-3)=-3^3-(-3)^2=18[/tex]
at s = -2
[tex]f(-2)=-2^3-(-2)^2\\\Rightarrow f(-2)=-12[/tex]
at s = -1
[tex]f(-1)=-1^3-(-1)^2\\\Rightarrow f(-1)=-2[/tex]
at s = 0
[tex]f(0)=0^3-0^2\\\Rightarrow f(0)=0[/tex]
at s = 1
[tex]f(1)=1^3-1^2\\\Rightarrow f(1)=0[/tex]
at s = 2
[tex]f(2)=2^3-2^2\\\Rightarrow f(2)=4[/tex]
at s = 3
[tex]f(3)=3^3-3^2\\\Rightarrow f(3)=18[/tex]
First difference
[tex]f(-2')=f(-2)-f(-3)=-12-18=-30[/tex]
[tex]f(-1')=f(-1)-f(-2')=-2--12=10[/tex]
[tex]f(0')=f(0)-f(-1)=0--2=2[/tex]
[tex]f(1')=f(1)-f(0)=0-0=0[/tex]
[tex]f(2')=f(2)-f(1)=4-0=4[/tex]
[tex]f(3')=f(3)-f(2)=18-4=14[/tex]
Second difference
[tex]f(-1'')=f(-1')-f(-2')=10--30=40[/tex]
[tex]f(0'')=f(0')-f(-1')=-8-40=-48[/tex]
[tex]f(1'')=f(1')-f(0')=-2--8=6[/tex]
[tex]f(2'')=f(2')-f(1')=14-4=10[/tex]
Third difference
[tex]f(0'')-f(-1'')=-8-40=48[/tex]
[tex]f(1'')-f(0'')=-2--8=6[/tex]
[tex]f(2'')-f(1'')=4--2=6[/tex]
[tex]f(2'')-f(1'')=10-4=6[/tex]
18 -12 -2 0 0 4 18
-30 10 2 0 4 14
40 -8 -2 4 10
-48 6 6 6
