展開する 1/((1-a/x)^3)-1/((1+a/x)^2)

解

展開する

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

\frac{5 a x ^{4} - 2 a ^{2} x ^{3} + a ^{3} x ^{2}}{x ^{5} - a x ^{4} - 2 a ^{2} x ^{3} + 2 a ^{3} x ^{2} + a ^{4} x - a ^{5}}

解答ステップ

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

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

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

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

因数

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

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

因数

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

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

以下の最小公倍数:

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

LCMに基づいて分数を調整する

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

分母が等しいので, 分数を組み合わせる:

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

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

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

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

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

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

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

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

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

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

拡張

(x^{3} - 3 x^{2} a + 3 x a^{2} - a^{3}) (x^{2} + 2 x a + a^{2}) : x^{5} - a x^{4} - 2 a^{2} x^{3} + 2 a^{3} x^{2} + a^{4} x - a^{5}

= \frac{x ^{3} ( x ^{2} + 2 a x + a ^{2} ) - x ^{2} ( x ^{3} - 3 a x ^{2} + 3 a ^{2} x - a ^{3} )}{x ^{5} - a x ^{4} - 2 a ^{2} x ^{3} + 2 a ^{3} x ^{2} + a ^{4} x - a ^{5}}

拡張

x^{3} (x^{2} + 2 x a + a^{2}) - x^{2} (x^{3} - 3 x^{2} a + 3 x a^{2} - a^{3}) : 5 a x^{4} - 2 a^{2} x^{3} + a^{3} x^{2}

= \frac{5 a x ^{4} - 2 a ^{2} x ^{3} + a ^{3} x ^{2}}{x ^{5} - a x ^{4} - 2 a ^{2} x ^{3} + 2 a ^{3} x ^{2} + a ^{4} x - a ^{5}}