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

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

Solve for x (complex solution)

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

Solution Steps

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

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

Cancel out -1 on both sides.

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

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}+xy+y^{2} and combine like terms.

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

Add y^{3} to 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}+xy+y^{2} 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.

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

Cancel out -1 on both sides.

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

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}+xy+y^{2} and combine like terms.

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

Add y^{3} to 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.

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