Question

(x-y-z)^2+(-x+y-z)^2+(x+y+z)

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

 

Answer 1

\(\left(x-y-z\right)^2+\left(-x+y-z\right)^2+\left(x+y+z\right)\\ =x^2+y^2+z^2-2xy+2yz-2xz+x^2+y^2+z^2-2xy-2yz+2xz+x+y+z\\ =x^2+y^2+z^2-4xy+x+y+z\)

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

Đặt \(x=a+b;y=b+c;z=a+c\), biểu thức trở thành

\(x^3+y^3+z^3-3xyz\\ =\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\\ =\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2\right)-3xy\left(x+y+z\right)\\ =\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)

Thay vào biểu thức, ta được

\(\left(a+b+b+c+c+a\right)\left[\left(a+b\right)^2+\left(b+c\right)^2+\left(c+a\right)^2-\left(a+b\right)\left(b+c\right)-\left(b+c\right)\left(c+a\right)-\left(a+b\right)\left(a+c\right)\right]\\ =2\left(a+b+c\right)\left(a^2+b^2+c^2-ab-ac-bc\right)\)