Question

verify that x3+y3+z3-3xyz=1/2(x+y+z)[(x-y)2+(y-z)2+(z-x)2]

Answer 1

First We take R.H.S & use the Formula [( a-b)²= a²+b²-2ab] & simplify it then R.H.S becomes equal to L.H.S


R.H.S

⇒ 1/2×(x + y + z) (x²+ y²-2xy +y²+ z²-2yz+x²+z²-2xz)

[( a-b)²= a²+b²-2ab]

⇒ 1/2×(x + y + z) (2x²+ 2y²+2z²-2xy -2yz-2xz)

⇒ 1/2×(x + y + z) 2(x² + y²+ z² – xy – yz – xz)

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

= x³+y³+z³-3xyz
= L.H.S

We know that,

[x³+ y³ + z³– 3xyz = (x + y + z)(x²+ y² + z² – xy – yz – xz)]

L.H.S = R.H.S

[x³+ y³ + z³– 3xyz = (x + y + z)(x²+ y² + z² – xy – yz – xz)]

================================================================

Hope this will help you....

Answer 2

Answer:

R.H.S=1/2(x+y+z)(x-y)^2(y-z)^2(z-x)^2

(by using identity(a-b)^2=a^2-2ab+b^2)

1/2(x+y+z)(x^2-2xy+b^2)(y^2-2yz+b^2)(z^2-2zx+x^2)

Now, adding the like terms

1/2(x+y+z)(2x^2+2y^2+2z^2-2xy-2yz-2zx)

Taking 2 as a common factor

1/2(x+y+z)2(x^2+y^2+z^2-xy-yz-zy)

1/2/2(x+y+z)(x^2+y^2+z^2-xy-yz-yz)

(x+y+z)(x^2+y^2+z^2-xy-yz-zx)

L.H.S=x^3+y^3+z^3-3xyz

we know that,

x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)

L.H.S=R.H.S

Hence, proved