Verify that x 3 + y 3 + z 3 - 3 x y z = 1 2 ( x + y + z ) ( x - y ) 2 + ( y - z ) 2 + ( z - x ) 2

We shall use the identity ( a - b ) 2 = a 2 + b 2 - 2 a b to verify the given equation x 3 + y 3 + z 3 - 3 x y z = 1 2 ( x + y + z ) ( x - y ) 2 + ( y - z ) 2 + ( z - x ) 2 We have, RHS = 1 2 ( x + y + z ) ( x - y ) 2 + ( y - z ) 2 + ( z - x ) 2 = 1 2 ( x + y + z ) x 2 + y 2 - 2 x y + y 2 + z 2 - 2 y z + z 2 + x 2 - 2 x z = 1 2 ( x + y + z ) 2 x 2 + 2 y 2 + 2 z 2 - 2 x y - 2 y z - 2 x z = 1 2 ( x + y + z ) 2 × x 2 + y 2 + z 2 - x y - y z - x z = ( x + y + z ) x 2 + y 2 + z 2 - x y - y z - x z = x 3 + y 3 + z 3 - 3 x y z So, LHS = RHS Hence, proved.

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MATHEMATICS>Chapter 2 - Polynomials>NCERT Exercises with Solutions

A K Upadhyay and S Upadhyay Solutions for Chapter: Polynomials, Exercise 7: NCERT Exercises with Solutions

Author:A K Upadhyay & S Upadhyay

A K Upadhyay Mathematics Solutions for Exercise - A K Upadhyay and S Upadhyay Solutions for Chapter: Polynomials, Exercise 7: NCERT Exercises with Solutions

Attempt the free practice questions on Chapter 2: Polynomials, Exercise 7: NCERT Exercises with Solutions with hints and solutions to strengthen your understanding. MATHEMATICS solutions are prepared by Experienced Embibe Experts.