jmc

Question

Let x, y, z be nonzero real numbers such that x + y + z = 0, and xy + xz + yz 0. Find all possible values of x^5 + y^5 + z^5xyz (xy + xz + yz).Enter all possible values, separated by commas.

Accepted Answer

Substituting z = -x - y, we get x^5 + y^5 - (x + y)^5xy(-x - y)(xy - x(x + y) - y(x + y)).Expanding the numerator and denominator, we get align* -5x^4 y + 10x^3 y^2 + 10x^2 y^3 + 5xy^4xy(x + y)(x^2 + xy + y^2) &= -5xy (x^3 + 2x^2 y + 2xy^2 + y^3)xy(x + y)(x^2 + xy + y^2) &= -5 (x^3 + 2x^2 y + 2xy^2 + y^3)(x + y)(x^2 + xy + y^2) &= -5 (x + y)(x^2 + xy + y^2)(x + y)(x^2 + xy + y^2) &= -5. align*Hence, the only possible value of the expression is -5.