To submit Let K be any field, and let a1,...,an be pairwise distinct elements of K (that is, a; + a; for all i + j). For each i = 1,...,n, define Pi = (x – a1) .· (x – aj–1)(x– aj+1)·…· (x– an) E...

Extracted text: To submit Let K be any field, and let a1,...,an be pairwise distinct elements of K (that is, a; + a; for all i + j). For each i = 1,...,n, define Pi = (x – a1) .· (x – aj–1)(x– aj+1)·…· (x– an) E K[x]. ... ... Note that the (x – a;) factor has been left out of pi, so deg p; =n– 1. (a) Prove that p;(a;) # 0 if and only if i = j. (b) Let b1,...,bk be elements of K (some of them maybe equal). Using part (a), explain how to find a polynomial q E K[x], with deg q < n="" (or="" q="0)," such="" that="" q(a;)="b;" for="" each="" i="1,.,n." [you="" don't="" have="" to="" include="" a="" proof.="" hint:="" think="" about="" the="" addition="" fact="" from="" the="" week="" 8="" submission="" question.]="" (c)="" prove="" that="" there="" cannot="" exist="" two="" different="" polynomials="" q,r="" e="" k[x],="" 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="">