Question

28. Do as directed: Verify that
x³+y³+z³-3xyz = (x+y+z) [ (x-y)² + (y-z)² + (z-x)²]​

Answer 1

Ans:- x³+y³+z³-3xyz = (x+y+z) [ (x-y)² + (y-z)²+ (z-x)²].

Now,

=> R.H.S = 1/2(x+y+z) [ (x-y)² + (y-z)² + (z-x)²]

=> R.H.S = 1/2(x+y+z)(x²+y²-2xy+y²+z²-2yz+z²+x²- 2xz).

=> R.H.S = 1/2(x+y+z) (2x²+2y²+2z²-2xy-2yz-2zx) => R.H.S = 1/2×2(x+y+z)(x²+y²+z²-xy-yz-zx)

=> R.H.S =x(x²+ y ²+ z² – xy – yz – zx) + y(x² + y² + z² – xy – yz – zx) + z(x² + y² + z² – xy – yz – zx) = x³ + xy² + xz ² – x²y – xyz – zx ² + x²y + y³ + yz ² – xy ² – y²z – xyz + x²z + y²z + z³ – xyz– yz² – xz²

= x³ + y ³ + z ³ – 3xyz (on simplification)

L.H.S = x³+y³+z³-3xyz

here is your answer

hope hope it will help you