展開する 3/(x+2)-2/(3x-1)-4/((3x-1)^2)

解

展開する

\frac{3}{x + 2} - \frac{2}{3 x - 1} - \frac{4}{( 3 x - 1 ) ^{2}}

\frac{21 x ^{2} - 32 x - 1}{9 x ^{3} + 12 x ^{2} - 11 x + 2}

解答ステップ

\frac{3}{x + 2} - \frac{2}{3 x - 1} - \frac{4}{( 3 x - 1 ) ^{2}}

(3 x - 1)^{2} = 9 x^{2} - 6 x + 1

= \frac{3}{x + 2} - \frac{2}{3 x - 1} - \frac{4}{9 x ^{2} - 6 x + 1}

因数

9 x^{2} - 6 x + 1 : (3 x - 1)^{2}

= \frac{3}{x + 2} - \frac{2}{3 x - 1} - \frac{4}{( 3 x - 1 ) ^{2}}

以下の最小公倍数:

x + 2, 3 x - 1, (3 x - 1)^{2} : (x + 2) (3 x - 1)^{2}

LCMに基づいて分数を調整する

= \frac{3 ( 3 x - 1 ) ^{2}}{( x + 2 ) ( 3 x - 1 ) ^{2}} - \frac{2 ( x + 2 ) ( 3 x - 1 )}{( x + 2 ) ( 3 x - 1 ) ^{2}} - \frac{4 ( x + 2 )}{( 3 x - 1 ) ^{2} ( 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{3 ( 3 x - 1 ) ^{2} - 2 ( x + 2 ) ( 3 x - 1 ) - 4 ( x + 2 )}{( x + 2 ) ( 3 x - 1 ) ^{2}}

(3 x - 1)^{2} = 9 x^{2} - 6 x + 1

= \frac{3 ( 3 x - 1 ) ^{2} - 2 ( x + 2 ) ( 3 x - 1 ) - 4 ( x + 2 )}{( x + 2 ) ( 9 x ^{2} - 6 x + 1 )}

(3 x - 1)^{2} = 9 x^{2} - 6 x + 1

= \frac{3 ( 9 x ^{2} - 6 x + 1 ) - 2 ( x + 2 ) ( 3 x - 1 ) - 4 ( x + 2 )}{( x + 2 ) ( 9 x ^{2} - 6 x + 1 )}

拡張

(x + 2) (9 x^{2} - 6 x + 1) : 9 x^{3} + 12 x^{2} - 11 x + 2

= \frac{3 ( 9 x ^{2} - 6 x + 1 ) - 2 ( x + 2 ) ( 3 x - 1 ) - 4 ( x + 2 )}{9 x ^{3} + 12 x ^{2} - 11 x + 2}

拡張

3 (9 x^{2} - 6 x + 1) - 2 (x + 2) (3 x - 1) - 4 (x + 2) : 21 x^{2} - 32 x - 1

= \frac{21 x ^{2} - 32 x - 1}{9 x ^{3} + 12 x ^{2} - 11 x + 2}