Good polynomials
Source: Problem 2 from ZIMO 2008
January 21, 2008
algebrapolynomialfactorizationsum of cubesalgebra proposed
Problem Statement
A polynomial with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable ) with integer coefficients.For example,the polynomials x^3 \minus{} 1 and 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3 are good.
a)Is the polynomial P(x) \equal{} 3x \plus{} 3x^7 good?
b)Is the polynomial P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008} good?
Justify your answers.