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– j-1)(x – a;+1)--- (x – an) E Kļx]. Note that...

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

Extracted text: 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– j-1)(x – a;+1)--- (x – an) E Kļx]. Note that the (x- ;) 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,...,b 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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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– j-1)(x – a;+1)--- (x – an) E Kļx]. Note that...

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