Math, asked by SrakshiRoy, 11 months ago

answer it correctly I will mark you as brainliest.

verify :
x² + y² + z³ - 3 xyz = 1/2(x + y + z) [(x-y)² + (y-z)² + (z-x) ² ]​

Answers

Answered by Anonymous
43

Given

x² + y² + z³ - 3 xyz = 1/2(x + y + z) [(x-y)² + (y-z)² + (z-x)²]

To Prove

L.H.S. = R.H.S.

{ x² + y² + z³ - 3 xyz = 1/2(x + y + z) [(x-y)² + (y-z)² + (z-x)²] }

\rule{200}2

Taking R.H.S.

⇒ 1/2(x + y + z) [(x-y)² + (y-z)² + (z-x)²]

⇒ 1/2 (x + y + z) [(x² + y² - 2xy) + (y² + z² - 2yz) + (z² + x² - 2zx)]

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

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

Take 2 as common

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

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

⇒ x³ - x²y + xy² - xyz + xz² - x²z + xy² - xy² + y³ - y²z + yz² - xyz + x²z - xyz + yz² - yz² - xz² + z³

After subtracting and adding we get,

⇒ x³ + y³ + z³ - 3xyz

L.H.S. = R.H.S.


Rythm14: noice :o
Anonymous: theku :p
Answered by RvChaudharY50
29

✯✯ To Prove ✯✯

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

|| ✰✰ ANSWER ✰✰ ||

Taking RHS we get :-

→ 1/2(x + y + z) [(x-y)² + (y-z)² + (z-x)²]

Using (a - b)² = + - 2ab now, we get,

1/2(x + y + z) [(x² + y² - 2xy) + (y² + z² - 2yz) + (z² + x² - 2zx) ]

→ 1/2(x + y + z) [x² + x² + y² + y² + z² + z² - 2xy - 2yz - 2zx ]

→ 1/2(x + y + z) [2x² + 2y² + 2z² - 2xy - 2yz - 2zx ]

Taking 2 common From Bracket now,

1/2(x + y + z) * 2 [x² + y² + z² - xy - yz - zx ]

→ (x + y + z)[x² + y² + z² - xy - yz - zx ]

Now, Either we use formula x³ + y³ + z³ - 3 xyz = (x + y + z)[x² + y² + z² - xy - yz - zx ] Directly. we will get our Result.

But Lets Try to Prove This also Now :-

→ (x + y + z)[x² + y² + z² - xy - yz - zx ]

→ x[x² + y² + z² - xy - yz - zx ] + y[x² + y² + z² - xy - yz - zx ] + z[x² + y² + z² - xy - yz - zx ]

→ [x³ + xy² + xz² - x²y - xyz - zx²] + [yx² + y³ + yz² - xy² - y²z - zxy ] + [zx² + zy² + z³ - xyz - yz² - z²x ]

→ x³ + xy² + xz² - x²y - xyz - zx² + yx² + y³ + yz² - xy² - y²z - zxy + zx² + zy² + z³ - xyz - yz² - z²x

→ x³ + y³ + z³ - xyz - xyz - xyz

→ x³ + y³ + z³ - 3xyz. = LHS.

✪✪ Hence Verified. ✪✪

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