(x+x_1)(x+x_2)(x-x_3)=x^3+(x_1+x_2+x_3)x^2+(x_1x_2+x_1 x_3+x_2x_3)x+x_1x_2x_3(x+x1)(x+x2)(x−x3)=x3+(x1+x2+x3)x2+(x1x2+x1x3+x2x3)x+x1x2x3 so
{(x_1+x_2+x_3=a),(x_1x_2+x_1 x_3+x_2x_3=b),(x_1x_2x_3=c):}
We are looking for a polynomial x^3+alpha x^2+beta x+gamma such that
{(x_1^3+x_2^3+x_3^3=alpha),(x_1^3x_2^3+x_1^3 x_3^3+x_2^3x_3^3=beta),(x_1^3x_2^3x_3^3=gamma):}
Clearly we have
gamma = c^3
now (x_1+x_2+x_3)^3 = x_1^3+x_2^3+x_3^3+3(x_1^2(x_2+x_3)+x_2^2(x_1+x_3)+x_3^2(x_1+x_2))+6 x_1x_2x_3
but
x_1^2(x_2+x_3)+x_2^2(x_1+x_3)+x_3^2(x_1+x_2)=(x_1+x_2+x_3)(x_1x_2+x_1 x_3+x_2x_3)+3x_1x_2x_3
so we have
a^3=x_1^3+x_2^3+x_3^3+3(ab+3c)+6c
and then
alpha = a^3-3(ab+3c)-6c
now making
(x_1x_2+x_1 x_3+x_2x_3)^3=x_1^3x_2^3+x_1^3 x_3^3+x_2^3x_3^3+3 (x_1^3 x_2^2 x_3 + x_1^2 x_2^3 x_3 + x_1^3 x_2 x_3^2 + x_1 x_2^3 x_3^2 + x_1^2 x_2 x_3^3 + x_1 x_2^2 x_3^3)+6x_1^2x_2^2x_3^2
but
x_1^3 x_2^2 x_3 + x_1^2 x_2^3 x_3 + x_1^3 x_2 x_3^2 + x_1 x_2^3 x_3^2 + x_1^2 x_2 x_3^3 + x_1 x_2^2 x_3^3=x_1x_2x_2(x_1^2(x_2+x_3)+x_2^2(x_1+x_3)+x_3^2(x_1+x_2))
so finally
b^3=beta+9c(ab+3c)+6c^3
and then
beta = b^3-9c(ab+3c)-6c^3 so the sought polynomial is
x^3+(a^3-3(ab+3c)-6c)x^2+(b^3-9c(ab+3c)-6c^3)x+c^3
