展開する 1/(x+10)+1/(x+5)+1/(x+1)-1/(x+2)

解

展開する

\frac{1}{x + 10} + \frac{1}{x + 5} + \frac{1}{x + 1} - \frac{1}{x + 2}

\frac{2 x ^{3} + 22 x ^{2} + 64 x + 80}{x ^{4} + 18 x ^{3} + 97 x ^{2} + 180 x + 100}

解答ステップ

\frac{1}{x + 10} + \frac{1}{x + 5} + \frac{1}{x + 1} - \frac{1}{x + 2}

以下の最小公倍数:

x + 10, x + 5, x + 1, x + 2 : (x + 5) (x + 1) (x + 2) (x + 10)

LCMに基づいて分数を調整する

= \frac{( x + 5 ) ( x + 1 ) ( x + 2 )}{( x + 5 ) ( x + 1 ) ( x + 2 ) ( x + 10 )} + \frac{( x + 1 ) ( x + 2 ) ( x + 10 )}{( x + 5 ) ( x + 1 ) ( x + 2 ) ( x + 10 )} + \frac{( x + 5 ) ( x + 2 ) ( x + 10 )}{( x + 5 ) ( x + 1 ) ( x + 2 ) ( x + 10 )} - \frac{( x + 5 ) ( x + 1 ) ( x + 10 )}{( x + 5 ) ( x + 1 ) ( x + 2 ) ( x + 10 )}

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 + 5 ) ( x + 1 ) ( x + 2 ) + ( x + 1 ) ( x + 2 ) ( x + 10 ) + ( x + 5 ) ( x + 2 ) ( x + 10 ) - ( x + 5 ) ( x + 1 ) ( x + 10 )}{( x + 5 ) ( x + 1 ) ( x + 2 ) ( x + 10 )}

拡張

(x + 5) (x + 1) (x + 2) (x + 10) : x^{4} + 18 x^{3} + 97 x^{2} + 180 x + 100

= \frac{( x + 5 ) ( x + 1 ) ( x + 2 ) + ( x + 1 ) ( x + 2 ) ( x + 10 ) + ( x + 5 ) ( x + 2 ) ( x + 10 ) - ( x + 5 ) ( x + 1 ) ( x + 10 )}{x ^{4} + 18 x ^{3} + 97 x ^{2} + 180 x + 100}

拡張

(x + 5) (x + 1) (x + 2) + (x + 1) (x + 2) (x + 10) + (x + 5) (x + 2) (x + 10) - (x + 5) (x + 1) (x + 10) : 2 x^{3} + 22 x^{2} + 64 x + 80

= \frac{2 x ^{3} + 22 x ^{2} + 64 x + 80}{x ^{4} + 18 x ^{3} + 97 x ^{2} + 180 x + 100}