Đáp án:
`k, (a + b + c)^3 - (a + b - c)^3 - (a - b + c)^3 - (-a + b + c)^3`
Đặt `a + b - c = x, a - b + c = y, -a + b + c = z`
`⇒ x + y + z = a + b - c + a - b + c - a + b + c = a + b + c`
Ta được:
`(x + y + z)^3 - x^3 - y^3 - z^3`
`= [(x + y) + z]^3 - x^3 - y^3 - z^3`
`= (x + y)^3 + 3(x + y)^2 z + 3(x + y)z^2 + z^3 - x^3 - y^3 - z^3`
`= x^3 + 3x^2 y + 3xy^2 + y^3 + 3(x^2 + 2xy + y^2).z + 3xz^2 + 3yz^2 - x^3 - y^3`
`= 3x^2 y + 3xy^2 + 3x^2 z + 6xyz + 3y^2 z + 3xz^2 + 3yz^2`
`= 3(x^2 y + xy^2 + x^2 z + 2xyz + y^2 z + xz^2 + yz^2)`
`= 3[(x^2 y + 2xyz + yz^2) + (xy^2 + y^2 z) + (x^2 z + xz^2)]`
`= 3[y(x^2 + 2xz + z^2) + y^2(x + z) + xz(x + z)]`
`= 3[y(x + z)^2 + y^2(x + z) + xz(x + z)]`
`= 3(x + z)[y(x + z) + y^2 + xz]`
`= 3(x + z)(xy + yz + y^2 + xz)`
`= 3(x + z)[(xy + y^2) + (yz + xz)]`
`= 3(x + z)[y(x + y) + z(y + x)]`
`= 3(x + z)(x + y)(y + z)`
`= 3[(a + b - c) + (-a + b + c)][(a + b - c) + (a - b + c)][(a - b + c) + (- a + b + c)]`
`= 3(a + b - c -a + b + c)(a + b - c + a - b + c)(a - b + c - a + b + c)`
`= 3.2b.2a.2c`
`= 24abc`
