Question

4. Verify that: x3 + y + 23 - 3xyz = 5(x + y + ) [(x - y)2 + (y – z)2 + (z – x)21

Please answer this question ​

Answer 1

Answer:

Verify that x ^ 3 + y ^ 3 + z ^ 3 - 3xyz = 1/2 * (x + y + z) * [(x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2]

Solving R.H.S

(x+y+z +y+z)[(x-y)2 + (y-2)² + (z-x)²]

ing * (a - b) ^ 2 = a ^ 2 + b ^ 2 - 2ab

=(x+y+z) [ (x² + y²-2xy) + (y² + z²2yz) + (2²+x²–2zx)]

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

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

= (x+y+z) [x² + y² + z²-xy-yz-zx]

We know x ^ 3 + y ^ 3 + z ^ 3 - 3xyz = (x + y + z)(x ^ 2 + y ^ 2 + z ^ 2 - xy - yz - zx)

= x³ + y³ +2³-3xyz

= L.H.S