Resol 3beta/alpha+1+3alpha/beta+1

$$\frac { 3 \beta } { \alpha + 1 } + \frac { 3 \alpha } { \beta + 1 }$$

Calcula

$$\frac{3\left(\alpha ^{2}+\alpha +\beta ^{2}+\beta \right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$

Passos de solució senzills

Per afegir o restar les expressions, amplieu-les perquè els denominadors coincideixin. El mínim comú múltiple de \alpha +1 i \beta +1 és \left(\alpha +1\right)\left(\beta +1\right). Multipliqueu \frac{3\beta }{\alpha +1} per \frac{\beta +1}{\beta +1}. Multipliqueu \frac{3\alpha }{\beta +1} per \frac{\alpha +1}{\alpha +1}.

$$\frac{3\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}+\frac{3\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$

Com que \frac{3\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} i \frac{3\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} tenen el mateix denominador, afegiu-los mitjançant l'addició dels seus numeradors.

$$\frac{3\beta \left(\beta +1\right)+3\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$

Feu les multiplicacions a 3\beta \left(\beta +1\right)+3\alpha \left(\alpha +1\right).

$$\frac{3\beta ^{2}+3\beta +3\alpha ^{2}+3\alpha }{\left(\alpha +1\right)\left(\beta +1\right)}$$

Expandiu \left(\alpha +1\right)\left(\beta +1\right).

$$\frac{3\beta ^{2}+3\beta +3\alpha ^{2}+3\alpha }{\alpha \beta +\alpha +\beta +1}$$

Factoritzar

$$\frac{3\left(\alpha ^{2}+\alpha +\beta ^{2}+\beta \right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$