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 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. (a) Prove that p;(a;) #0 if and only if i = j. (b) Let bi,...,bz 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
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Question regarding algebra
![Let K be any field, and let a1,...,a, be pairwise distinct elements of K (that is, a; +a;
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.
(a) Prove that p:(a;) #0 if and only if i = j.
(b) Let b1,...,bz be elements of K (some of them maybe equal). Using part (a),
explain how to find a polynomial q E K[x], with degq <n (or q = 0), such that
q(a;) = b; for each i = 1,...,n.
You don't have to include a proof.
(c) Prove that there cannot exist two different polynomials q,r EK[x], both of degree
less than n, such that q(a;) = r(a;) for each i = 1,...,n.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ac52ece-fa98-4f63-b4a0-88c320f6c626%2F335341cc-3669-4803-9587-a48f304884e5%2Fn416jkw_processed.jpeg&w=3840&q=75)
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