it Let K be any field, and let a1,.…,an be pairwise distinct elements of K (that is, a; † aj for all i+ j). For each i= 1, ...,n, define Pi3 (х — а).-(х - а-1) (х— ан1).. (х— аn) € К\x). Note that the...

C) please $it Let K be any field, and let a1,.…,an be pairwise distinct elements of K (that is, a; † aj for all i+ j). For each i= 1, ...,n, define Pi3 (х — а).-(х - а-1) (х— ан1).. (х— аn) € К\x). Note that the (x– a;) factor has been left out of pi, so deg pi = 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 sheets.] $

Extracted text: it Let K be any field, and let a1,.…,an be pairwise distinct elements of K (that is, a; † aj for all i+ j). For each i= 1, ...,n, define Pi3 (х — а).-(х - а-1) (х— ан1).. (х— аn) € К\x). Note that the (x– a;) factor has been left out of pi, so deg pi = 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

Jun 05, 2022

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it Let K be any field, and let a1,.…,an be pairwise distinct elements of K (that is, a; † aj for all i+ j). For each i= 1, ...,n, define Pi3 (х — а).-(х - а-1) (х— ан1).. (х— аn) € К\x). Note that the...

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