a) (x - 2)(x2 - 2x + 4)(x - 2)( x2 + 2x + 4)
= (x - 2)2(x - 2)2(x + 2)2
= (x - 2)4(x + 2)2
b) (a + b + c)3 - (b + c - a)3 - (a - b + c)3 - (a + b - c)3
Đặt a+b-c=x, c+a-b=y, b+c-a=z
=>x+y+z=a+b-c+c+a-b+b+c-a=a+b+c
Ta có hằng đẳng thức:
(x+y+z)^3-3x-3y-3z=3(x+y)(x+z)(y+z)
=>(a+b+c)^3-(b+c-a)^3-(a+c-b)^3-(a+b-c)^3=(x+y+z)^3-x^3-y^3-z^3
=3(x+y)(x+z)(y+z)
=3(a+b-c+c+a-b)(c+a-b+b+c-a)(b+c-a+a+b-c)
=3.2a.2b.2c
=24abc
c) (a + b)3 + (b + c)3 + (c + a)3 - 3(a + b)(b + c)(c + a)
Đặt x = a+b; y = b+c; z = c+a ta có:
x3+y3+z3−3xyz
= (x+y)3−3xy(x−y)+z3−3xyz
=[(x+y)3+z3]−3xy(x+y+z)
=(x+y+z)3−3z(x+y)(x+y+z)−3xy(x−y−z)
=(x+y+z)[(x+y+z)2−3z(x+y)−3xy]
=(x+y+z)(x2+y2+z2+2xy+2xz+2yz−3xz−3yz−3xy)
=(x+y+z)(x2+y2+z2−xy−yz−yx)
Thay vào ta có:
(a+b+b+c+c+a)[(a+b)2+(b+c)2+(c+a)2−(a+b)(b+c)−(b+c)(c+a)−(c+a)(a+b)]
=(2a+2b+2c)(a2−ab−ac+b2−bc+c2)
=2(a+b+c)(a2−ab−ac+b2−bc+c2)
