Without the use of pexiderized versions of abstract polynomials
theory, we show that on 2-divisible groups the functional equation
f(x + y) + g(x + y) + g(x − y) = f(x) + f(y)+ 2g(x) + 2g(y)
forces the unknown functions f and g to be additive and quadratic, respectively,
modulo a constant.
Motivated by the observation that the equation
f(x + y) + f(x2) = f(x) + f(y)+ f(x)2
implies both the additivity and multiplicativity of f, we deal also with the alienation
phenomenon of equations in a single and several variables.