因数 1/(x-1)-(3x+3)/(x^2+x-2)+1/(x+2)

解

因数

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

- \frac{1}{x - 1}

解答ステップ

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

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

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

以下の最小公倍数:

x - 1, (x - 1) (x + 2), x + 2 : (x - 1) (x + 2)

LCMに基づいて分数を調整する

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

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 + 2 - 3 ( x + 1 ) + x - 1}{( x - 1 ) ( x + 2 )}

x + 2 - 3 (x + 1) + x - 1 = 2 x - 3 (x + 1) + 2 - 1

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

拡張

2 x - 3 (x + 1) + 2 - 1 : - x - 2

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

- x - 2 = - (x + 2)

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

改良

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

共通因数を約分する:

x + 2

= - \frac{1}{x - 1}