(x^(3)-1)/(x^(2)+y+z)+(y^(3)-1)/(y^(2)+x+z)+(z^(3)-1)/(z^(2)+y+x)

asked by guest
on Sep 21, 2024 at 10:56 am



You asked:

Evaluate the expression: \(\frac{{x}^{3} - 1}{{x}^{2} + y + z} + \frac{{y}^{3} - 1}{{y}^{2} + x + z} + \frac{{z}^{3} - 1}{{z}^{2} + y + x}\)

MathBot Answer:

Evaluated



\(\displaystyle \frac{{x}^{3} - 1}{{x}^{2} + y + z} + \frac{{y}^{3} - 1}{{y}^{2} + x + z} + \frac{{z}^{3} - 1}{{z}^{2} + y + x} = \frac{x^{3} - 1}{x^{2} + y + z} + \frac{y^{3} - 1}{x + y^{2} + z} + \frac{z^{3} - 1}{x + y + z^{2}} \)

Expanded

\[\frac{{x}^{3} - 1}{{x}^{2} + y + z} + \frac{{y}^{3} - 1}{{y}^{2} + x + z} + \frac{{z}^{3} - 1}{{z}^{2} + y + x} = \frac{x^{3}}{x^{2} + y + z} + \frac{y^{3}}{x + y^{2} + z} + \frac{z^{3}}{x + y + z^{2}} - \frac{1}{x + y + z^{2}} - \frac{1}{x + y^{2} + z} - \frac{1}{x^{2} + y + z}\]

Factored

\[\frac{{x}^{3} - 1}{{x}^{2} + y + z} + \frac{{y}^{3} - 1}{{y}^{2} + x + z} + \frac{{z}^{3} - 1}{{z}^{2} + y + x} = \frac{x^{5} + x^{4} y^{2} + x^{4} y + x^{4} z^{2} + x^{4} z + 2 x^{3} y^{3} + x^{3} y^{2} z^{2} + x^{3} y z + 2 x^{3} z^{3} - 2 x^{3} + x^{2} y^{4} + x^{2} y^{3} z^{2} + x^{2} y^{2} z^{3} - x^{2} y^{2} - x^{2} y + x^{2} z^{4} - x^{2} z^{2} - x^{2} z - x^{2} + y^{4} x + y^{3} x z - y^{2} x + z^{3} x y - 3 x y + z^{4} x - z^{2} x - 3 x z + y^{5} + y^{4} z^{2} + y^{4} z + 2 y^{3} z^{3} - 2 y^{3} + y^{2} z^{4} - y^{2} z^{2} - y^{2} z - y^{2} + z^{4} y - z^{2} y - 3 y z + z^{5} - 2 z^{3} - z^{2}}{\left(x^{2} + y + z\right) \left(x + y^{2} + z\right) \left(x + y + z^{2}\right)}\]

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