expandir ((8-x^2-y^2)^3)/3

Solução

expandir

\frac{( 8 - x ^{2} - y ^{2} ) ^{3}}{3}

- \frac{x ^{6}}{3} - x^{4} y^{2} + 8 x^{4} - x^{2} y^{4} + 16 x^{2} y^{2} - 64 x^{2} - \frac{y ^{6}}{3} + 8 y^{4} - 64 y^{2} + \frac{512}{3}

Passos da solução

\frac{( 8 - x ^{2} - y ^{2} ) ^{3}}{3}

(8 - x^{2} - y^{2})^{3} = - x^{6} - 3 x^{4} y^{2} + 24 x^{4} - 3 x^{2} y^{4} + 48 x^{2} y^{2} - 192 x^{2} - y^{6} + 24 y^{4} - 192 y^{2} + 512

= \frac{( - x ^{6} - 3 x ^{4} y ^{2} + 24 x ^{4} - 3 x ^{2} y ^{4} + 48 x ^{2} y ^{2} - 192 x ^{2} - y ^{6} + 24 y ^{4} - 192 y ^{2} + 512 )}{3}

Remover os parênteses:

(- a) = - a

= \frac{- x ^{6} - 3 x ^{4} y ^{2} + 24 x ^{4} - 3 x ^{2} y ^{4} + 48 x ^{2} y ^{2} - 192 x ^{2} - y ^{6} + 24 y ^{4} - 192 y ^{2} + 512}{3}

Aplicar as propriedades das frações:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- x ^{6} - 3 x ^{4} y ^{2} + 24 x ^{4} - 3 x ^{2} y ^{4} + 48 x ^{2} y ^{2} - 192 x ^{2} - y ^{6} + 24 y ^{4} - 192 y ^{2} + 512}{3} = - \frac{x ^{6}}{3} - \frac{3 x ^{4} y ^{2}}{3} + \frac{24 x ^{4}}{3} - \frac{3 x ^{2} y ^{4}}{3} + \frac{48 x ^{2} y ^{2}}{3} - \frac{192 x ^{2}}{3} - \frac{y ^{6}}{3} + \frac{24 y ^{4}}{3} - \frac{192 y ^{2}}{3} + \frac{512}{3}

= - \frac{x ^{6}}{3} - \frac{3 x ^{4} y ^{2}}{3} + \frac{24 x ^{4}}{3} - \frac{3 x ^{2} y ^{4}}{3} + \frac{48 x ^{2} y ^{2}}{3} - \frac{192 x ^{2}}{3} - \frac{y ^{6}}{3} + \frac{24 y ^{4}}{3} - \frac{192 y ^{2}}{3} + \frac{512}{3}

\frac{3 x ^{4} y ^{2}}{3} = x^{4} y^{2}

\frac{24 x ^{4}}{3} = 8 x^{4}

\frac{3 x ^{2} y ^{4}}{3} = x^{2} y^{4}

\frac{48 x ^{2} y ^{2}}{3} = 16 x^{2} y^{2}

\frac{192 x ^{2}}{3} = 64 x^{2}

\frac{24 y ^{4}}{3} = 8 y^{4}

\frac{192 y ^{2}}{3} = 64 y^{2}

= - \frac{x ^{6}}{3} - x^{4} y^{2} + 8 x^{4} - x^{2} y^{4} + 16 x^{2} y^{2} - 64 x^{2} - \frac{y ^{6}}{3} + 8 y^{4} - 64 y^{2} + \frac{512}{3}

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s im pl i f y 1 0^{10} δ (x - 9)

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