(b) Let b1,...,br 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
(b) Let b1,...,br 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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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B) 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.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b3f8b07-7556-48e1-a197-c9780c4ecda8%2F44f663f1-c8bd-4bfb-98c7-82e18c20151b%2Fty49je_processed.jpeg&w=3840&q=75)
Transcribed Image 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 <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.]
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