Let be a polynomial of degree less than or equal to 2. Find all that satisfy the given conditions.
- , .
Since is a polynomial of degree less than or equal to 2 we have
for constants for all of the below.
Let , then is of degree at most 2, and we have three points at which (since implies ). By part (d) of the previous exercise we have all of the coefficients of are 0 and for all . Thus,
Then, with ,
Finally, with and , we have,
Again, from we have,
Then, with and we have,
Just substituting these values we have,
Where are arbitrary real numbers.