Resolver x+1+3x/x^2-x-6

$$x + 1 + \frac { 3 x } { x ^ { 2 } - x - 6 }$$

Calcular

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

Pasos de solución curta

Factoriza x^{2}-x-6.

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

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

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

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

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

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

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

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

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

Expande \left(x-3\right)\left(x+2\right).

$$\frac{x^{3}-4x-6}{x^{2}-x-6}$$

Diferenciar w.r.t. x

$$\frac{x^{4}-2x^{3}-14x^{2}+12x+18}{\left(\left(x-3\right)\left(x+2\right)\right)^{2}}$$