Let P3(R) be the set of all polynomials of degree 3 or less with real coefficients. Prove that P3(R) is a vector space when considering + as point-wise sum and · as point-wise scalar multiplication (that is, if p1, P2 E P3(R), then pi +p2 is the polynomial such that p1 + P2(x) = p1(x) + p2(x), while a · p1 is the polynomial such that a · p1(x) = ap1(x)). Can you find a finite subset of P3(R) that spans P3 (R)?

Let P3(R) be the set of all polynomials of degree 3 or less with real coefficients. Prove that P3(R) is a vector space when considering + as point-wise sum and · as point-wise scalar multiplication (that is, if p1, P2 E P3(R), then pi +p2 is the polynomial such that p1 + P2(x) = p1(x) + p2(x), while a · p1 is the polynomial such that a · p1(x) = ap1(x)). Can you find a finite subset of P3(R) that spans P3 (R)?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 4E: Write each of the following polynomials as a products of its leading coefficient and a finite number...

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