The polynomial is...
Kind regards
Question: If are distinct numbers, find a polynomial of degree which is at and 0 at for . Hint: the product of all for is zero at if . .
Confusion: The part of the question that I don't understand is the 'which is at and 0 at for .' What are and ? Does it mean that for any two numbers we choose from the sequence , we get as 0, for one, and 1, for the other? And how would you go about finding this polynomial?
The problem asks to find, for every , a polynomial with the given property, which is a total of polynomials.
In the denominator, you have a product of numbers. In the numerator, you have a product of expressions of the form for various constants .For example, it's not apparent to me that it's of degree .
This is similar to the interpolation polynomial in the Lagrange form (the link uses notation instead of ).