Answer:
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Verify that x³ + y³ + z³ - 3xyz = 1/2 (x + y + z)[(x - y)² + (y - z)² + (z - x)²]
Solution:
Taking the RHS and evaluating,
R.H.S. = 1/2 ( x + y + z) [(x - y)² + ( y - z)² + (z - x)²]
= 1/2 ( x + y + z) [(x² - 2xy + y²) + ( y² - 2yz + z²) + (z² - 2zx + x²)]
= 1/2 ( x + y + z) [2x² + 2y² + 2z² - 2xy - 2yz - 2zx]
= 1/2 ( x + y + z) (2) [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 - x²z + x²y + y³ + yz² - xy² - y²z - xyz + zx² + y²z + z³ - xyz - yz² - xz²
On simplifying,
= x³ + y³ + z³ - 3xyz = LHS
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Verified answer
Answer:
cuemath_logo
Verify that x³ + y³ + z³ - 3xyz = 1/2 (x + y + z)[(x - y)² + (y - z)² + (z - x)²]
Solution:
Taking the RHS and evaluating,
R.H.S. = 1/2 ( x + y + z) [(x - y)² + ( y - z)² + (z - x)²]
= 1/2 ( x + y + z) [(x² - 2xy + y²) + ( y² - 2yz + z²) + (z² - 2zx + x²)]
= 1/2 ( x + y + z) [2x² + 2y² + 2z² - 2xy - 2yz - 2zx]
= 1/2 ( x + y + z) (2) [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 - x²z + x²y + y³ + yz² - xy² - y²z - xyz + zx² + y²z + z³ - xyz - yz² - xz²
On simplifying,
= x³ + y³ + z³ - 3xyz = LHS