Verified answer

a) (a+b+c)^3 - (a+b-c)^3 - (b+c-a)^3 - (c+a-b)^3 (*)

đặt:

x = a + b - c

y = b + c - a

z = c + a - b

=> x + y+ z = a + b + c

và:

x + y = 2b

y + z = 2c

x + z = 2a

(*) thành:

(x + y + z)^3 - x^3 - y^3 - z^3

= [(x + y + z) - z][(x+ y + z)^2 + x^2 + x(x+ y + z)] - (y + z)(y^2 + z^2 - yz)

= (y+z)(x^2 + y^2 + z^2 + 2xy + 2yz + 2xz + 2x^2 + xy + xz) - (y + z)(y^2 + z^2 - yz)

= (y+z)(x^2 + y^2 + z^2 + 2xy + 2yz + 2xz + 2x^2 + xy + xz - y^2 - z^2 + yz)

= (y+z)(3x^2 + 3xy + 3yz + 3xz )

= 3(y+z)(x^2 + xy + yz + xz )

= 3(y+z)[x(x+y) + z(x+y)]

= 3(x+y)(y+z)(x+z)

= 3(2c)(2b)(2a) = 24abc

b) (x-y)^3+(y-z)^3+(z-x)^3 (1)

x - y = a

y - z = b

z - x = c

=> a + b + c = 0

(1) thành:

a^3 + b^3 + c^3 = (a + b)^3 - 3ab(a + b) + c^3

= (a + b + c)^3 - 3(a+b).c(a + b + c) - 3ab(a + b)

= - 3abc

hay: (x - y)^3 + (y - z)^3 + (z - x)^3 = 3(x - y)(y - z)(x - z)