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

Step-by-step 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