Question

Rút gọn:

B=\(\left(a+b+c\right)^3+\left(a-b-c\right)^3+\left(b-c-a\right)^3+\left(c-a-b\right)^3\)

Answer 1

Lời giải:

Đặt \((a-b-c,b-c-a,c-a-b)=(x,y,z)\)

\(\Rightarrow x+y+z=-(a+b+c)\)

\(\Rightarrow -(x+y+z)=a+b+c\). Khi đó:

\(B=[-(x+y+z)]^3+x^3+y^3+z^3\)

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

\(=x^3+y^3+z^3-[(x+y)^3+3(x+y)z(x+y+z)+z^3]\)

\(=x^3+y^3+z^3-[x^3+3xy(x+y)+y^3+3(x+y)z(x+y+z)+z^3]\)

\(=-3xy(x+y)-3(x+y)z(x+y+z)\)

\(=-3(x+y)[xy+z(x+y+z)]\)

\(=-3(x+y)(z+x)(z+y)=-3(-2c)(-2b)(-2a)=24abc\)