jmc

Question

Suppose that x, y, and z satisfy the equations align* xyz &= 4, x^3 + y^3 + z^3 &= 4, xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z &= 12. align*Calculate the value of xy + yz + zx.

Accepted Answer

Let s_1 = x + y + z and s_2 = xy + xz + yz. Then align* s_1 s_2 &= (x + y + z)(xy + xz + yz) &= x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2 + 3xyz &= 12 + 3 4 = 24. align*Also, align* s_1^3 &= (x + y + z)^3 &= (x^3 + y^3 + z^3) + 3(x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2) + 6xyz &= 4 + 3 12 + 6 4 = 64, align*so s_1 = 4. Hence, s_2 = 24s_1 = 6.