Prove that there cannot exist two different polynomials q,r E K, both of degree less than n. such that g(a;) = r(a;) for eachi=1,... ,n.
Prove that there cannot exist two different polynomials q,r E K, both of degree less than n. such that g(a;) = r(a;) for eachi=1,... ,n.
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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Question
![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.
|](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2687b65-8ce5-496e-92b7-0b0b0eaf2c3f%2F86876e31-a171-4c06-ad5f-e73822983a2e%2Fu84otzf_processed.jpeg&w=3840&q=75)
Transcribed Image 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.
|
![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.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2687b65-8ce5-496e-92b7-0b0b0eaf2c3f%2F86876e31-a171-4c06-ad5f-e73822983a2e%2Fszi0y3c_processed.jpeg&w=3840&q=75)
Transcribed Image 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.]
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