Solucionar 2/x^3-4x^2-3x+18-3x/5x^2-15

$$\frac { 2 } { x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18 } - \frac { 3 x } { 5 x ^ { 2 } - 15 }$$

Calcular

$$\frac{-3x^{4}+12x^{3}+19x^{2}-54x-30}{5\left(x+2\right)\left(x-3\right)^{2}\left(x^{2}-3\right)}$$

Solución en pasos breves

Factorice x^{3}-4x^{2}-3x+18. Factorice 5x^{2}-15.

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

Para sumar o restar expresiones, expándalas para que sus denominadores sean iguales. El mínimo común múltiplo de \left(x+2\right)\left(x-3\right)^{2} y 5\left(x^{2}-3\right) es 5\left(x+2\right)\left(x-3\right)^{2}\left(x^{2}-3\right). Multiplica \frac{2}{\left(x+2\right)\left(x-3\right)^{2}} por \frac{5\left(x^{2}-3\right)}{5\left(x^{2}-3\right)}. Multiplica \frac{3x}{5\left(x^{2}-3\right)} por \frac{\left(x+2\right)\left(x-3\right)^{2}}{\left(x+2\right)\left(x-3\right)^{2}}.

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

Como \frac{2\times 5\left(x^{2}-3\right)}{5\left(x+2\right)\left(x-3\right)^{2}\left(x^{2}-3\right)} y \frac{3x\left(x+2\right)\left(x-3\right)^{2}}{5\left(x+2\right)\left(x-3\right)^{2}\left(x^{2}-3\right)} tienen el mismo denominador, reste sus numeradores para restarlos.

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

Haga las multiplicaciones en 2\times 5\left(x^{2}-3\right)-3x\left(x+2\right)\left(x-3\right)^{2}.

$$\frac{10x^{2}-30-3x^{4}+18x^{3}-27x^{2}-6x^{3}+36x^{2}-54x}{5\left(x+2\right)\left(x-3\right)^{2}\left(x^{2}-3\right)}$$

Combine los términos semejantes en 10x^{2}-30-3x^{4}+18x^{3}-27x^{2}-6x^{3}+36x^{2}-54x.

$$\frac{19x^{2}-30-3x^{4}+12x^{3}-54x}{5\left(x+2\right)\left(x-3\right)^{2}\left(x^{2}-3\right)}$$

Expande 5\left(x+2\right)\left(x-3\right)^{2}\left(x^{2}-3\right).

$$\frac{19x^{2}-30-3x^{4}+12x^{3}-54x}{5x^{5}-20x^{4}-30x^{3}+150x^{2}+45x-270}$$

Factorizar

$$\frac{-3x^{4}+12x^{3}+19x^{2}-54x-30}{5\left(x+2\right)\left(x-3\right)^{2}\left(x^{2}-3\right)}$$