Verify that x 3 + y 3 + z 3 – 3xyz = \(\frac{1}{2}\) (x + y + z) [(x - y) 2 + (y - z) 2 + (z - x) 2 ].

It is known that, x 3 + y 3 + z 3 - 3xyz = (x + y + z) (x 2 + y 2 + z 2 - xy - yz - zx) = \(\frac{1}{2}\) (x + y + z) [2x 2 + 2y 2 + 2z 2 - 2xy - 2yz - 2zx] = \(\frac{1}{2}\) (x + y + z) [(x 2 + y 2 - 2xy) + (y 2 + z 2 - 2yz) + (x 2 + z 2 -2zx)] = \(\frac{1}{2}\) (x + y + z) [(x - y) 2 + (y - z) 2 + (z - x) 2 ]

Question:

Verify that x³ + y³ + z³ – 3xyz = \(\frac{1}{2}\) (x + y + z) [(x - y)² + (y - z)² + (z - x)²].

CBSE Class IX

Updated On: Nov 18, 2023

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Solution and Explanation

It is known that,

x³ + y³ + z³ - 3xyz

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

= \(\frac{1}{2}\) (x + y + z) [2x² + 2y² + 2z² - 2xy - 2yz - 2zx]

=\(\frac{1}{2}\) (x + y + z) [(x² + y² - 2xy) + (y² + z² - 2yz) + (x² + z² -2zx)]

= \(\frac{1}{2}\) (x + y + z) [(x - y)² + (y - z)² + (z - x)²]

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Verify that x 3 + y 3 + z 3 – 3xyz = \(\frac{1}{2}\) (x + y + z) [(x - y)2 + (y - z)2 + (z - x)2].

Solution and Explanation

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Questions Asked in CBSE Class IX exam

Verify that x³ + y³ + z³ – 3xyz = \(\frac{1}{2}\) (x + y + z) [(x - y)² + (y - z)² + (z - x)²].