簡約 (2a+3c)/(2a+c)-(2b-3a)/(3a+b)-(2c(3a+b))/(6a^2+2ab+3ac+bc)

解

簡約

\frac{2 a + 3 c}{2 a + c} - \frac{2 b - 3 a}{3 a + b} - \frac{2 c ( 3 a + b )}{6 a ^{2} + 2 ab + 3 a c + b c}

\frac{72 a ^{4} + 12 a ^{3} b + 72 a ^{3} c - 4 a ^{2} b ^{2} + 36 a ^{2} c ^{2} - c ^{2} b ^{2} - 18 c ^{2} a ^{2} + 12 a ^{2} c b - 4 a c b ^{2} + 3 a c ^{2} b}{( 2 a + c ) ( 3 a + b ) ( 6 a ^{2} + 2 ab + 3 a c + c b )}

解答ステップ

\frac{2 a + 3 c}{2 a + c} - \frac{2 b - 3 a}{3 a + b} - \frac{2 c ( 3 a + b )}{6 a ^{2} + 2 ab + 3 a c + b c}

以下の最小公倍数:

2 a + c, 3 a + b, 6 a^{2} + 2 ab + 3 a c + b c : (2 a + c) (3 a + b) (6 a^{2} + 2 ab + 3 a c + c b)

LCMに基づいて分数を調整する

= \frac{( 2 a + 3 c ) ( 3 a + b ) ( 6 a ^{2} + 2 ab + 3 a c + c b )}{( 2 a + c ) ( 3 a + b ) ( 6 a ^{2} + 2 ab + 3 a c + c b )} - \frac{( 2 b - 3 a ) ( 2 a + c ) ( 6 a ^{2} + 2 ab + 3 a c + c b )}{( 2 a + c ) ( 3 a + b ) ( 6 a ^{2} + 2 ab + 3 a c + c b )} - \frac{2 c ( 2 a + c ) ( 3 a + b ) ^{2}}{( 2 a + c ) ( 3 a + b ) ( 6 a ^{2} + 2 ab + 3 a c + c b )}

分数の規則を適用する:

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

= \frac{( 2 a + 3 c ) ( 3 a + b ) ( 6 a ^{2} + 2 ab + 3 a c + c b ) - ( 2 b - 3 a ) ( 2 a + c ) ( 6 a ^{2} + 2 ab + 3 a c + c b ) - 2 c ( 2 a + c ) ( 3 a + b ) ^{2}}{( 2 a + c ) ( 3 a + b ) ( 6 a ^{2} + 2 ab + 3 a c + c b )}

簡素化

(2 a + 3 c) (3 a + b) (6 a^{2} + 2 ab + 3 a c + c b) - (2 b - 3 a) (2 a + c) (6 a^{2} + 2 ab + 3 a c + c b) - 2 c (2 a + c) (3 a + b)^{2} : 72 a^{4} + 12 a^{3} b + 72 a^{3} c - 4 a^{2} b^{2} + 36 a^{2} c^{2} - c^{2} b^{2} - 18 c^{2} a^{2} + 12 a^{2} c b - 4 a c b^{2} + 3 a c^{2} b

= \frac{72 a ^{4} + 12 a ^{3} b + 72 a ^{3} c - 4 a ^{2} b ^{2} + 36 a ^{2} c ^{2} - c ^{2} b ^{2} - 18 c ^{2} a ^{2} + 12 a ^{2} c b - 4 a c b ^{2} + 3 a c ^{2} b}{( 2 a + c ) ( 3 a + b ) ( 6 a ^{2} + 2 ab + 3 a c + c b )}