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In this problem we prove, using linear algebra, that for any u, v, w there is a unique polynomial P(x) of degree at most 2 such that P(1) = u, P(2)= v, P(3): =w. Let P₁(x) = (x-2)(x-3) (-2) (1-2)(1-3) P2(x) (x-1)(x-3) (2-1)(2-3) and P3(x) 2) Prove that the list P₁, P2, P3 is a basis of P₂ (R). = (x-1)(x-2) (3-1)(3-2).

In this problem we prove, using linear algebra, that for any u, v, w there is a unique polynomial P(x) of degree at most 2 such that P(1) = u, P(2)= v, P(3): =w. Let P₁(x) = (x-2)(x-3) (-2) (1-2)(1-3) P2(x) (x-1)(x-3) (2-1)(2-3) and P3(x) 2) Prove that the list P₁, P2, P3 is a basis of P₂ (R). = (x-1)(x-2) (3-1)(3-2).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.2: Divisibility And Greatest Common Divisor
Problem 18E
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In this problem we prove, using linear algebra, that for any u, v, w there is a unique polynomial P(x) of degree at most 2 such that P(1) = u, P(2) = v, P(3) = w. Let P₁(x) = P₂(x) = (x-2)(x-3) (1-2)(1-3)* (x-1)(x-3) (2-1)(2-3) and P3(x) 2) Prove that the list P₁, P2, P3 is a basis of P₂ (R). - (x-1)(x-2) (3-1)(3-2)*
Transcribed Image Text:In this problem we prove, using linear algebra, that for any u, v, w there is a unique polynomial P(x) of degree at most 2 such that P(1) = u, P(2) = v, P(3) = w. Let P₁(x) = P₂(x) = (x-2)(x-3) (1-2)(1-3)* (x-1)(x-3) (2-1)(2-3) and P3(x) 2) Prove that the list P₁, P2, P3 is a basis of P₂ (R). - (x-1)(x-2) (3-1)(3-2)*
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