Đáp án:
`2(x^4 + y^4 + z^4) - (x^2 + y^2 + z^2)^2 - 2(x^2 + y^2 + z^2)(x + y + z)^2 + (x + y +z)^4`
Đặt `x^4 + y^4 + z^4 = a, x^2 + y^2 + z^2 = b, x + y + z = c`, đa thức trở thành:
`2a - b^2 - 2bc^2 + c^4`
`= 2a + b^2 - 2bc^2 + c^4 - 2b^2`
`= (2a - 2b^2) + [b^2 - 2.b.c^2 + (c^2)^2]`
`= 2(a - b^2) + (b - c^2)^2`
`= 2[x^4 + y^4 + z^4 - (x^2 + y^2 + z^2)^2] + [x^2 + y^2 + z^2 - (x + y + z)^2]^2`
`= 2[x^4 + y^4 + z^4 - (x^4 + y^4 + z^4 + 2x^2 y^2 + 2y^2 z^2 + 2z^2 x^2)] + [x^2 + y^2 + z^2 - (x^2 + y^2 + z^2 + 2xy + 2yz + 2xz)]^2`
`= 2[x^4 + y^4 + z^4 - x^4 - y^4 - z^4 - 2x^2 y^2 - 2y^2 z^2 - 2z^2 x^2] + [x^2 + y^2 + z^2 - x^2 - y^2 - z^2 - 2xy - 2yz - 2xz]^2`
`= 2(-2x^2 y^2 - 2y^2 z^2 - 2x^2 z^2) + (-2xy - 2yz - 2xz)^2`
`= 2.(-2)(x^2 y^2 + y^2 z^2 + x^2 z^2) + [-2(xy + yz + xz)]^2`
`= -4(x^2 y^2 + y^2 z^2 + x^2 z^2) + 4(x^2 y^2 + y^2 z^2 + x^2 z^2 + 2xy^2 z + 2xyz^2 + 2x^2 yz)`
`= -4(x^2 y^2 + y^2 z^2 + x^2 z^2 - x^2 y^2 - y^2 z^2 - x^2 z^2 - 2xy^2 z - 2xyz^2 - 2x^2 yz)`
`= -4(-2xy^2 z - 2xyz^2 - 2x^2 yz)`
`= (-4).(-2xyz)(y + z + x)`
`= 8xyz(x + y + z)`
* Áp dụng hằng đẳng thức : `(A + B + C)^2 = A^2 + B^2 + C^2 + 2AB + 2BC + 2AC`
