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5

Part of 1973 Spain Mathematical Olympiad

Problems(1)

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

Source: Spanish Mathematical Olympiad 1973 P5

12/23/2022
Consider the set of all polynomials of degree less than or equal to 44 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)31, x - 2, (x -2)^2, (x - 2)^3 and (x−2)4(x -2)^4 form a base of this space. c) Express the polynomial 7+2x−45x2+3x47 + 2x - 45x^2 + 3x^4 in the previous base.
algebrapolynomialvectorlinear algebra