Let K be any field, and let a1,..., An be pairwise distinet elements of K (that is, a,ta for all i j). For each i = 1,...,n, define Pi = (x-a1).. (x– a;-1)(x=a;+1)·…· (x- a,) E K[x]. Note that the...

Extracted text: Let K be any field, and let a1,..., An be pairwise distinet elements of K (that is, a,ta for all i j). For each i = 1,...,n, define Pi = (x-a1).. (x– a;-1)(x=a;+1)·…· (x- a,) E K[x]. Note that the (x-a) factor has been left out of p,, so deg P, =n-1. |


Extracted text: Prove that there cannot exist two different polynomials q, r€K, both of degree less than n, such that q(a,) = r(a,) for each i 1,...,n. [You may assume without proof facts from previous coursework sheets.]