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polynomials of degree<=4 with rational coefficients as vector space

Source: Spanish Mathematical Olympiad 1973 P5

December 23, 2022

algebrapolynomialvectorlinear algebra

Problem Statement

Consider the set of all polynomials of degree less than or equal to

4

with rational coefficients. a) Prove that it has a vector space structure over the field of numbers rational. b) Prove that the polynomials

1, x - 2, (x -2)^2, (x - 2)^3

and

(x -2)^4

form a base of this space. c) Express the polynomial

7 + 2x - 45x^2 + 3x^4

in the previous base.