| Let K be any field, and let a,...,dn 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,)...

| Let K be any field, and let a,...,dn 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-a;_1)(x- a;±1)·…(x – a,) E K[x].<br>Note that the (x-a,) factor has been left out of p, so deg p=n-1.<br>(a) Prove that p(aj)+0 if and only if i j.<br>....b, be elements of K (some of them maybe equal). Using part (a).<br>(b) Let bi,.<br>explain how to find a polynomial q E K, with deg q <n (or q= 0), such that<br>q(a1) = b, 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 EK, both of degree<br>less than n, such that q(a;) = r(a,) for each i=1,...,n.<br>[You may assume without proof facts from previous coursework sheets.]<br>

Extracted text: | Let K be any field, and let a,...,dn 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(aj)+0 if and only if i j. ....b, be elements of K (some of them maybe equal). Using part (a). (b) Let bi,. explain how to find a polynomial q E K, with deg q

Jun 04, 2022

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| Let K be any field, and let a,...,dn 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,)...

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