(x^4*(y-z)+y^4*(z-x)+z^4*(x-y))/(2*(x^2+y^2+z^2+xy+xz+zy))

asked by guest
on Jan 23, 2025 at 8:40 am



You asked:

Evaluate the expression: \(\frac{{x}^{4} \left(y - z\right) + {y}^{4} \left(z - x\right) + {z}^{4} \left(x - y\right)}{2 \left({x}^{2} + {y}^{2} + {z}^{2} + x y + x z + z y\right)}\)

MathBot Answer:

Evaluated



\(\displaystyle \frac{{x}^{4} \left(y - z\right) + {y}^{4} \left(z - x\right) + {z}^{4} \left(x - y\right)}{2 \left({x}^{2} + {y}^{2} + {z}^{2} + x y + x z + z y\right)} = \frac{x^{4} \left(y - z\right) + y^{4} \left(- x + z\right) + z^{4} \left(x - y\right)}{2 x^{2} + 2 x y + 2 x z + 2 y^{2} + 2 y z + 2 z^{2}} \)

Expanded

\[\frac{{x}^{4} \left(y - z\right) + {y}^{4} \left(z - x\right) + {z}^{4} \left(x - y\right)}{2 \left({x}^{2} + {y}^{2} + {z}^{2} + x y + x z + z y\right)} = \frac{x^{4} y}{2 x^{2} + 2 x y + 2 x z + 2 y^{2} + 2 y z + 2 z^{2}} - \frac{x^{4} z}{2 x^{2} + 2 x y + 2 x z + 2 y^{2} + 2 y z + 2 z^{2}} - \frac{y^{4} x}{2 x^{2} + 2 x y + 2 x z + 2 y^{2} + 2 y z + 2 z^{2}} + \frac{z^{4} x}{2 x^{2} + 2 x y + 2 x z + 2 y^{2} + 2 y z + 2 z^{2}} + \frac{y^{4} z}{2 x^{2} + 2 x y + 2 x z + 2 y^{2} + 2 y z + 2 z^{2}} - \frac{z^{4} y}{2 x^{2} + 2 x y + 2 x z + 2 y^{2} + 2 y z + 2 z^{2}}\]

Factored

\[\frac{{x}^{4} \left(y - z\right) + {y}^{4} \left(z - x\right) + {z}^{4} \left(x - y\right)}{2 \left({x}^{2} + {y}^{2} + {z}^{2} + x y + x z + z y\right)} = \frac{\left(x - y\right) \left(x - z\right) \left(y - z\right)}{2}\]

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