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
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] }
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.
✯✯ 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 = a² + b² - 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