展开 1/(x+10)+1/(x+5)+1/(x+1)-1/(x+2)

解答

展开

\frac{1}{x + 10} + \frac{1}{x + 5} + \frac{1}{x + 1} - \frac{1}{x + 2}

\frac{2 x ^{3} + 22 x ^{2} + 64 x + 80}{x ^{4} + 18 x ^{3} + 97 x ^{2} + 180 x + 100}

求解步骤

\frac{1}{x + 10} + \frac{1}{x + 5} + \frac{1}{x + 1} - \frac{1}{x + 2}

x + 10, x + 5, x + 1, x + 2

的最小公倍数:

(x + 5) (x + 1) (x + 2) (x + 10)

根据最小公倍数调整分式

= \frac{( x + 5 ) ( x + 1 ) ( x + 2 )}{( x + 5 ) ( x + 1 ) ( x + 2 ) ( x + 10 )} + \frac{( x + 1 ) ( x + 2 ) ( x + 10 )}{( x + 5 ) ( x + 1 ) ( x + 2 ) ( x + 10 )} + \frac{( x + 5 ) ( x + 2 ) ( x + 10 )}{( x + 5 ) ( x + 1 ) ( x + 2 ) ( x + 10 )} - \frac{( x + 5 ) ( x + 1 ) ( x + 10 )}{( x + 5 ) ( x + 1 ) ( x + 2 ) ( x + 10 )}

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{( x + 5 ) ( x + 1 ) ( x + 2 ) + ( x + 1 ) ( x + 2 ) ( x + 10 ) + ( x + 5 ) ( x + 2 ) ( x + 10 ) - ( x + 5 ) ( x + 1 ) ( x + 10 )}{( x + 5 ) ( x + 1 ) ( x + 2 ) ( x + 10 )}

乘开

(x + 5) (x + 1) (x + 2) (x + 10) : x^{4} + 18 x^{3} + 97 x^{2} + 180 x + 100

= \frac{( x + 5 ) ( x + 1 ) ( x + 2 ) + ( x + 1 ) ( x + 2 ) ( x + 10 ) + ( x + 5 ) ( x + 2 ) ( x + 10 ) - ( x + 5 ) ( x + 1 ) ( x + 10 )}{x ^{4} + 18 x ^{3} + 97 x ^{2} + 180 x + 100}

乘开

(x + 5) (x + 1) (x + 2) + (x + 1) (x + 2) (x + 10) + (x + 5) (x + 2) (x + 10) - (x + 5) (x + 1) (x + 10) : 2 x^{3} + 22 x^{2} + 64 x + 80

= \frac{2 x ^{3} + 22 x ^{2} + 64 x + 80}{x ^{4} + 18 x ^{3} + 97 x ^{2} + 180 x + 100}

e x p an d e^{2 (x + 1)}

e x p an d \frac{10}{( s + 1 ) ( s + 2 ) ( s + 5 )}

e x p an d y (y - (- 7) = - 8 (x - 6))

e x p an d d x ((2 x^{3} - 5) (2 x - 4))

e x p an d e^{y^{2}} \cdot (- y + 1)^{2}