簡約 (4a^2+3a+2)/(a^3-1)-(1-2a)/(a^2+a+1)

解

簡約

\frac{4 a ^{2} + 3 a + 2}{a ^{3} - 1} - \frac{1 - 2 a}{a ^{2} + a + 1}

\frac{6 a ^{2} + 3}{( a - 1 ) ( a ^{2} + a + 1 )}

解答ステップ

\frac{4 a ^{2} + 3 a + 2}{a ^{3} - 1} - \frac{1 - 2 a}{a ^{2} + a + 1}

以下の最小公倍数:

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

LCMに基づいて分数を調整する

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

分数の規則を適用する:

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

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

簡素化

4 a^{2} + 3 a + 2 - (1 - 2 a) (a - 1) : 6 a^{2} + 3

= \frac{6 a ^{2} + 3}{( a - 1 ) ( a ^{2} + a + 1 )}