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

R.H.S = (x + y + z)[(x - y) 2 + (y - z) 2 + (z - x) 2 ] = (x + y + z)[(x 2 + y 2 - 2xy) + (y 2 + z 2 - 2yz) + (z 2 + x 2 - 2zx)] [∵ (a - b) 2 = (a) 2 + (b) 2 - 2ab] = (x + y + z)[2x 2 + 2y 2 + 2z 2 - 2xy - 2yz - 2zx] = (x + y + z) 2[x 2 + y 2 + z 2 - xy - yz - zx] = (x + y + z)[x 2 + y 2 + z 2 - xy - yz - zx] = x 3 + y 3 +z 3 - 3xyz [∵ x 3 + y 3 +z 3 - 3xyz = (x + y + z)(x 2 + y 2 + z 2 - xy - yz - zx)] Hence, L.H.S = R.H.S

Factorise each of the following:
(i) 27y³ + 125z³
(ii) 64m³ - 343n³

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Factorise : 27x³ + y³ + z³ - 9xyz

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If x + y + z = 0, show that x³ + y³ + z³ = 3xyz

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Without actually calculating the cubes, find the value of each of the following:
(i) (-12)³ + (7)³ + (5)³
(ii) (28)³ + (-15)³ + (-13)³

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Mathematics

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

Polynomials

Related Questions

Factorise each of the following:
(i) 27y³ + 125z³
(ii) 64m³ - 343n³

Factorise : 27x³ + y³ + z³ - 9xyz

If x + y + z = 0, show that x³ + y³ + z³ = 3xyz

Without actually calculating the cubes, find the value of each of the following:
(i) (-12)³ + (7)³ + (5)³
(ii) (28)³ + (-15)³ + (-13)³

Mathematics

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

Polynomials

Related Questions

Factorise each of the following:(i) 27y3 + 125z3(ii) 64m3 - 343n3

Factorise : 27x3 + y3 + z3 - 9xyz

If x + y + z = 0, show that x3 + y3 + z3 = 3xyz

Without actually calculating the cubes, find the value of each of the following:(i) (-12)3 + (7)3 + (5)3(ii) (28)3 + (-15)3 + (-13)3

Factorise each of the following:
(i) 27y³ + 125z³
(ii) 64m³ - 343n³

Factorise : 27x³ + y³ + z³ - 9xyz

If x + y + z = 0, show that x³ + y³ + z³ = 3xyz

Without actually calculating the cubes, find the value of each of the following:
(i) (-12)³ + (7)³ + (5)³
(ii) (28)³ + (-15)³ + (-13)³