Question : Is there a cubic polynomial that takes these values
You substitute the x with the values your given.
Then you get an equation with each of the first 4 values, which gives you a system of 4 equations with 4 unknowns (a,b,c,d).
Then see if the remaining values are such that the equations are satisfied.
I suggest you first consider 0,1,-1 because they're nice values to deal with
You don't understand a thing we're saying, do you?
Is there a cubic polynomial that takes these values?
The general cubic polynomial is: .
. . and we must determine the four coefficients:
We can use the first four values of the function and create a system of equations.
Solve the system of equations: .
Hence, the cubic is: .
. . which goes through the first four points.
. . It does not go through the fifth point.
Therefore, there is no cubic polynomial through the given six points.
we only need four points to constuct a cubic polynomial so
I choose those points which are located at
To find the cubic polynomial which passes through those points
we need to find the first , second and the third difference ,
Therefore , by using Newton's Interpolation Formula ,
the cubic polynomial is :
To check whether there is a cubic polynomial passing through the other points either , we just sub.
and see if the values the polynomial gives us equal to and respectively .