Resolver 1/x^2-1-2/x^2+3x-4+1/x^2-2x-3

$$\frac { 1 } { x ^ { 2 } - 1 } - \frac { 2 } { x ^ { 2 } + 3 x - 4 } + \frac { 1 } { x ^ { 2 } - 2 x - 3 }$$

Calcular

$$\frac{2\left(4x-5\right)}{\left(x-3\right)\left(x+4\right)\left(x^{2}-1\right)}$$

Pasos de solución curta

Factoriza x^{2}-1. Factoriza x^{2}+3x-4.

$$\frac{1}{\left(x-1\right)\left(x+1\right)}-\frac{2}{\left(x-1\right)\left(x+4\right)}+\frac{1}{x^{2}-2x-3}$$

Para sumar ou restar expresións, expándeas para facer que os seus denominadores sexan iguais. O mínimo común múltiplo de \left(x-1\right)\left(x+1\right) e \left(x-1\right)\left(x+4\right) é \left(x-1\right)\left(x+1\right)\left(x+4\right). Multiplica \frac{1}{\left(x-1\right)\left(x+1\right)} por \frac{x+4}{x+4}. Multiplica \frac{2}{\left(x-1\right)\left(x+4\right)} por \frac{x+1}{x+1}.

$$\frac{x+4}{\left(x-1\right)\left(x+1\right)\left(x+4\right)}-\frac{2\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x+4\right)}+\frac{1}{x^{2}-2x-3}$$

Dado que \frac{x+4}{\left(x-1\right)\left(x+1\right)\left(x+4\right)} e \frac{2\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x+4\right)} teñen o mesmo denominador, réstaos mediante a resta dos seus numeradores.

$$\frac{x+4-2\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x+4\right)}+\frac{1}{x^{2}-2x-3}$$

Fai as multiplicacións en x+4-2\left(x+1\right).

$$\frac{x+4-2x-2}{\left(x-1\right)\left(x+1\right)\left(x+4\right)}+\frac{1}{x^{2}-2x-3}$$

Combina como termos en x+4-2x-2.

$$\frac{-x+2}{\left(x-1\right)\left(x+1\right)\left(x+4\right)}+\frac{1}{x^{2}-2x-3}$$

Factoriza x^{2}-2x-3.

$$\frac{-x+2}{\left(x-1\right)\left(x+1\right)\left(x+4\right)}+\frac{1}{\left(x-3\right)\left(x+1\right)}$$

Para sumar ou restar expresións, expándeas para facer que os seus denominadores sexan iguais. O mínimo común múltiplo de \left(x-1\right)\left(x+1\right)\left(x+4\right) e \left(x-3\right)\left(x+1\right) é \left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+4\right). Multiplica \frac{-x+2}{\left(x-1\right)\left(x+1\right)\left(x+4\right)} por \frac{x-3}{x-3}. Multiplica \frac{1}{\left(x-3\right)\left(x+1\right)} por \frac{\left(x-1\right)\left(x+4\right)}{\left(x-1\right)\left(x+4\right)}.

$$\frac{\left(-x+2\right)\left(x-3\right)}{\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+4\right)}+\frac{\left(x-1\right)\left(x+4\right)}{\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+4\right)}$$

Dado que \frac{\left(-x+2\right)\left(x-3\right)}{\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+4\right)} e \frac{\left(x-1\right)\left(x+4\right)}{\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+4\right)} teñen o mesmo denominador, súmaos mediante a suma dos seus numeradores.

$$\frac{\left(-x+2\right)\left(x-3\right)+\left(x-1\right)\left(x+4\right)}{\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+4\right)}$$

Fai as multiplicacións en \left(-x+2\right)\left(x-3\right)+\left(x-1\right)\left(x+4\right).

$$\frac{-x^{2}+3x+2x-6+x^{2}+4x-x-4}{\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+4\right)}$$

Combina como termos en -x^{2}+3x+2x-6+x^{2}+4x-x-4.

$$\frac{8x-10}{\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+4\right)}$$

Expande \left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+4\right).

$$\frac{8x-10}{x^{4}+x^{3}-13x^{2}-x+12}$$

Diferenciar w.r.t. x

$$\frac{2\left(43-130x+67x^{2}+12x^{3}-12x^{4}\right)}{\left(\left(x-3\right)\left(x+4\right)\left(x^{2}-1\right)\right)^{2}}$$