Solve (x+y)(x^2+y^2-xy)=x^3+y^3

$$(x+y)( { x }^{ 2 } + { y }^{ 2 } -xy)= { x }^{ 3 } + { y }^{ 3 }$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply x+y by x^{2}+y^{2}-xy and combine like terms.

$$x^{3}+y^{3}=x^{3}+y^{3}$$

Subtract x^{3} from both sides.

$$x^{3}+y^{3}-x^{3}=y^{3}$$

Combine x^{3} and -x^{3} to get 0.

$$y^{3}=y^{3}$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for y (complex solution)

$$y\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply x+y by x^{2}+y^{2}-xy and combine like terms.

$$x^{3}+y^{3}=x^{3}+y^{3}$$

Subtract y^{3} from both sides.

$$x^{3}+y^{3}-y^{3}=x^{3}$$

Combine y^{3} and -y^{3} to get 0.

$$x^{3}=x^{3}$$

Reorder the terms.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply x+y by x^{2}+y^{2}-xy and combine like terms.

$$x^{3}+y^{3}=x^{3}+y^{3}$$

Subtract x^{3} from both sides.

$$x^{3}+y^{3}-x^{3}=y^{3}$$

Combine x^{3} and -x^{3} to get 0.

$$y^{3}=y^{3}$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Solve for y

$$y\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply x+y by x^{2}+y^{2}-xy and combine like terms.

$$x^{3}+y^{3}=x^{3}+y^{3}$$

Subtract y^{3} from both sides.

$$x^{3}+y^{3}-y^{3}=x^{3}$$

Combine y^{3} and -y^{3} to get 0.

$$x^{3}=x^{3}$$

Reorder the terms.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{R}$$