Question

prove that (x+y)^3+(y+z)^3+(z+x)^3=3(x+y)(y+z)(z+x)​

Answer 1

Step-by-step explanation:

Let x+y=a, y + z = b and z+x=C

Then,

(x+y)³ + (y+z)³ + (z+x)³ −3(x+y)(y+z) (z + x)

a3 + b³ + c3-3abc

(a+b+c)(a² + b ² + c²-ab-bc-ca)

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

= 2(x+y+z)(x² + y² + 2xy + y² +z²+2yz+z²+x²+2xz-(xy + x2 + y² +yz)

-(yz + xy +z²+xz)-(xz+yz + x² + xy)]

= 2(x+y+z)(x² + y² + 2xy + y² +z²+2yz + z² + x² + 2xz-xy-xz-y²-yz-vz-xy-z²-xz-xz-vz-x²-xy)

= 2(x+y+z)(x² + y² + z² - xy-vz-zx)

2(x³+y³+23-3xyz)