Resolver 3beta/alpha+1+3alpha/beta+1

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

Avaliar

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

Passos Rápidos de Resolução

Para adicionar ou subtrair expressões, expanda-as para que os denominadores sejam iguais. O mínimo múltiplo comum de \alpha +1 e \beta +1 é \left(\alpha +1\right)\left(\beta +1\right). Multiplique \frac{3\beta }{\alpha +1} vezes \frac{\beta +1}{\beta +1}. Multiplique \frac{3\alpha }{\beta +1} vezes \frac{\alpha +1}{\alpha +1}.

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

Uma vez que \frac{3\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} e \frac{3\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} têm o mesmo denominador, some-os ao somar os respetivos numeradores.

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

Efetue as multiplicações em 3\beta \left(\beta +1\right)+3\alpha \left(\alpha +1\right).

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

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

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

Fatorizar

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