Solve alphabeta^2+alpha^2beta=alphabeta(alpha+beta)=

$$\alpha \beta ^ { 2 } + \alpha ^ { 2 } \beta = \alpha \beta ( \alpha + \beta ) =$$

Solve for α

$$\alpha \in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply \alpha \beta by \alpha +\beta .

$$\alpha \beta ^{2}+\alpha ^{2}\beta =\beta \alpha ^{2}+\alpha \beta ^{2}$$

Subtract \beta \alpha ^{2} from both sides.

$$\alpha \beta ^{2}+\alpha ^{2}\beta -\beta \alpha ^{2}=\alpha \beta ^{2}$$

Combine \alpha ^{2}\beta and -\beta \alpha ^{2} to get 0.

$$\alpha \beta ^{2}=\alpha \beta ^{2}$$

Subtract \alpha \beta ^{2} from both sides.

$$\alpha \beta ^{2}-\alpha \beta ^{2}=0$$

Combine \alpha \beta ^{2} and -\alpha \beta ^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any \alpha .

$$\alpha \in \mathrm{R}$$

Solve for β

$$\beta \in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply \alpha \beta by \alpha +\beta .

$$\alpha \beta ^{2}+\alpha ^{2}\beta =\beta \alpha ^{2}+\alpha \beta ^{2}$$

Subtract \beta \alpha ^{2} from both sides.

$$\alpha \beta ^{2}+\alpha ^{2}\beta -\beta \alpha ^{2}=\alpha \beta ^{2}$$

Combine \alpha ^{2}\beta and -\beta \alpha ^{2} to get 0.

$$\alpha \beta ^{2}=\alpha \beta ^{2}$$

Subtract \alpha \beta ^{2} from both sides.

$$\alpha \beta ^{2}-\alpha \beta ^{2}=0$$

Combine \alpha \beta ^{2} and -\alpha \beta ^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any \beta .

$$\beta \in \mathrm{R}$$