展开 3/(x+2)-2/(3x-1)-4/((3x-1)^2)

解答

展开

\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}

求解步骤

\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}

分解

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}}

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

的最小公倍数:

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

根据最小公倍数调整分式

= \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 )}

因为分母相等，所以合并分式:

\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 )}

乘开

(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}

乘开

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}

e x p an d \frac{( a ( x + h ) + b ) - ( a x + b )}{h}

e x p an d 50 (\frac{12 sin ( \frac{π \cdot x}{12} )}{π} + x)

\frac{d}{d x} ((x - y^{3})^{2} - x^{3})

e x p an d (y = x^{2} + 2 x) (x + 2)

e x p an d \frac{x ^{2} + 5 x - 1}{( x ^{2} + 3 ) ^{2}}