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受欢迎的代数 >

展开 1/((x-a)^3)-1/((x+a)^3)

解答

展开

\frac{1}{( x - a ) ^{3}} - \frac{1}{( x + a ) ^{3}}

解答

\frac{6 a x ^{2} + 2 a ^{3}}{x ^{6} - 3 a ^{2} x ^{4} + 3 a ^{4} x ^{2} - a ^{6}}

求解步骤

\frac{1}{( x - a ) ^{3}} - \frac{1}{( x + a ) ^{3}}

(x - a)^{3} = x^{3} - 3 a x^{2} + 3 a^{2} x - a^{3}

= \frac{1}{x ^{3} - 3 a x ^{2} + 3 a ^{2} x - a ^{3}} - \frac{1}{( x + a ) ^{3}}

(x + a)^{3} = x^{3} + 3 a x^{2} + 3 a^{2} x + a^{3}

= \frac{1}{x ^{3} - 3 a x ^{2} + 3 a ^{2} x - a ^{3}} - \frac{1}{x ^{3} + 3 a x ^{2} + 3 a ^{2} x + a ^{3}}

分解

x^{3} - 3 x^{2} a + 3 x a^{2} - a^{3} : (x - a)^{3}

= \frac{1}{( x - a ) ^{3}} - \frac{1}{x ^{3} + 3 x ^{2} a + 3 x a ^{2} + a ^{3}}

分解

x^{3} + 3 x^{2} a + 3 x a^{2} + a^{3} : (x + a)^{3}

= \frac{1}{( x - a ) ^{3}} - \frac{1}{( x + a ) ^{3}}

(x - a)^{3}, (x + a)^{3}

的最小公倍数:

(x - a)^{3} (x + a)^{3}

根据最小公倍数调整分式

= \frac{( x + a ) ^{3}}{( x - a ) ^{3} ( x + a ) ^{3}} - \frac{( x - a ) ^{3}}{( x - a ) ^{3} ( x + a ) ^{3}}

因为分母相等，所以合并分式:

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

= \frac{( x + a ) ^{3} - ( x - a ) ^{3}}{( x - a ) ^{3} ( x + a ) ^{3}}

(x - a)^{3} = x^{3} - 3 a x^{2} + 3 a^{2} x - a^{3}

= \frac{( x + a ) ^{3} - ( x - a ) ^{3}}{( x ^{3} - 3 a x ^{2} + 3 a ^{2} x - a ^{3} ) ( x + a ) ^{3}}

(x + a)^{3} = x^{3} + 3 a x^{2} + 3 a^{2} x + a^{3}

= \frac{( x + a ) ^{3} - ( x - a ) ^{3}}{( x ^{3} - 3 a x ^{2} + 3 a ^{2} x - a ^{3} ) ( x ^{3} + 3 a x ^{2} + 3 a ^{2} x + a ^{3} )}

乘开

(x^{3} - 3 x^{2} a + 3 x a^{2} - a^{3}) (x^{3} + 3 x^{2} a + 3 x a^{2} + a^{3}) : x^{6} - 3 a^{2} x^{4} + 3 a^{4} x^{2} - a^{6}

= \frac{( x + a ) ^{3} - ( x - a ) ^{3}}{x ^{6} - 3 a ^{2} x ^{4} + 3 a ^{4} x ^{2} - a ^{6}}

乘开

(x + a)^{3} - (x - a)^{3} : 6 a x^{2} + 2 a^{3}

= \frac{6 a x ^{2} + 2 a ^{3}}{x ^{6} - 3 a ^{2} x ^{4} + 3 a ^{4} x ^{2} - a ^{6}}

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