Solve (m-n)(m^2+mn+n^2)=m^3-n^3

$$( m - n ) ( m ^ { 2 } + m n + n ^ { 2 } ) = m ^ { 3 } - n ^ { 3 }$$

Solve for m (complex solution)

$$m\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply m-n by m^{2}+mn+n^{2} and combine like terms.

$$m^{3}-n^{3}=m^{3}-n^{3}$$

Subtract m^{3} from both sides.

$$m^{3}-n^{3}-m^{3}=-n^{3}$$

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

$$-n^{3}=-n^{3}$$

Cancel out -1 on both sides.

$$n^{3}=n^{3}$$

Reorder the terms.

$$\text{true}$$

This is true for any m.

$$m\in \mathrm{C}$$

Solve for n (complex solution)

$$n\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply m-n by m^{2}+mn+n^{2} and combine like terms.

$$m^{3}-n^{3}=m^{3}-n^{3}$$

Add n^{3} to both sides.

$$m^{3}-n^{3}+n^{3}=m^{3}$$

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

$$m^{3}=m^{3}$$

Reorder the terms.

$$\text{true}$$

This is true for any n.

$$n\in \mathrm{C}$$

Solve for m

$$m\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply m-n by m^{2}+mn+n^{2} and combine like terms.

$$m^{3}-n^{3}=m^{3}-n^{3}$$

Subtract m^{3} from both sides.

$$m^{3}-n^{3}-m^{3}=-n^{3}$$

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

$$-n^{3}=-n^{3}$$

Cancel out -1 on both sides.

$$n^{3}=n^{3}$$

Reorder the terms.

$$\text{true}$$

This is true for any m.

$$m\in \mathrm{R}$$

Solve for n

$$n\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply m-n by m^{2}+mn+n^{2} and combine like terms.

$$m^{3}-n^{3}=m^{3}-n^{3}$$

Add n^{3} to both sides.

$$m^{3}-n^{3}+n^{3}=m^{3}$$

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

$$m^{3}=m^{3}$$

Reorder the terms.

$$\text{true}$$

This is true for any n.

$$n\in \mathrm{R}$$