簡約 (a^2-4ac+3bc)/(a^2-ab+bc-ac)+(a+3b)/(b-a)+(a+2c)/(a-c)

解

簡約

\frac{a ^{2} - 4 a c + 3 b c}{a ^{2} - ab + b c - a c} + \frac{a + 3 b}{b - a} + \frac{a + 2 c}{a - c}

\frac{- a ^{4} + 5 a ^{3} b - 4 a ^{2} b ^{2} + 2 a ^{3} c - a ^{2} c ^{2} - 3 b ^{2} c ^{2} - c ^{2} b ^{2} - 10 a ^{2} c b + 5 a c ^{2} b + 8 a c b ^{2}}{( b - a ) ( a - c ) ( a ^{2} - ab - a c + c b )}

解答ステップ

\frac{a ^{2} - 4 a c + 3 b c}{a ^{2} - ab + b c - a c} + \frac{a + 3 b}{b - a} + \frac{a + 2 c}{a - c}

以下の最小公倍数:

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

LCMに基づいて分数を調整する

= \frac{( a ^{2} - 4 a c + 3 b c ) ( b - a ) ( a - c )}{( b - a ) ( a - c ) ( a ^{2} - ab - a c + c b )} + \frac{( a + 3 b ) ( a - c ) ( a ^{2} - ab - a c + c b )}{( b - a ) ( a - c ) ( a ^{2} - ab - a c + c b )} + \frac{( a + 2 c ) ( b - a ) ( a ^{2} - ab - a c + c b )}{( b - a ) ( a - c ) ( a ^{2} - ab - 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{( a ^{2} - 4 a c + 3 b c ) ( b - a ) ( a - c ) + ( a + 3 b ) ( a - c ) ( a ^{2} - ab - a c + c b ) + ( a + 2 c ) ( b - a ) ( a ^{2} - ab - a c + c b )}{( b - a ) ( a - c ) ( a ^{2} - ab - a c + c b )}

簡素化

(a^{2} - 4 a c + 3 b c) (b - a) (a - c) + (a + 3 b) (a - c) (a^{2} - ab - a c + c b) + (a + 2 c) (b - a) (a^{2} - ab - a c + c b) : - a^{4} + 5 a^{3} b - 4 a^{2} b^{2} + 2 a^{3} c - a^{2} c^{2} - 3 b^{2} c^{2} - c^{2} b^{2} - 10 a^{2} c b + 5 a c^{2} b + 8 a c b^{2}

= \frac{- a ^{4} + 5 a ^{3} b - 4 a ^{2} b ^{2} + 2 a ^{3} c - a ^{2} c ^{2} - 3 b ^{2} c ^{2} - c ^{2} b ^{2} - 10 a ^{2} c b + 5 a c ^{2} b + 8 a c b ^{2}}{( b - a ) ( a - c ) ( a ^{2} - ab - a c + c b )}