Question

[Quang Huy Điền]

CMR : nếu x + y + z = 0 thì :

\(2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\)

Answer 1

[Trương Hồng Hạnh]

Ta có: x + y + z = 0 <=> y + z = -x

(y+z)⁵ = (-x)⁵

y⁵ + z⁵ + 5y⁴z + 10y³z² + 10y²z³ + 5yz⁴ = -x⁵

y⁵ + z⁵ + 5y⁴z + 10y³z² + 10y²z³ + 5yz⁴ + x⁵ = 0

x⁵ + y⁵ + z⁵ +5xyz[ y³ + 2y²z + 2yz² + z³ ] = 0

x⁵ + y⁵ + z⁵ + 5xyz[(y+z)(y² -yz -z²)+ 2yz(x+z)] = 0

x⁵ + y⁵ + z⁵ +5xyz[(y+z)(y² +yz + z²)] = 0

2.(x⁵ + y⁵ + z⁵) + 5xyz(y+z)(y²+yz+z²) - (x⁵ + y⁵ + z⁵) = 0

2(x⁵ + y⁵ + z⁵) - 5xyz[(y²+2yz+z²)+y²+z²] = 0

2(x⁵ + y⁵ + z⁵) = 5xyz[(y+z)² + y² + z²]

2(x⁵ + y⁵ + z⁵) = 5xyz[(-x)² + y² + z²]

2(x⁵ + y⁵ + z⁵) = 5xyz(x² + y² + z²).