Verify: <br/><br/>(i) $x^{3}+y^{3}=(x+y)\left(x^{2}-x y+y^{2}\right)$<br/><br/>(ii) $x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)$

Solution:

(i) It is known that,

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

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

$=(x+y)\left[(x+y)^{2}-3 x y\right]$

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

$=(x+y)\left(x^{2}+y^{2}-x y\right)$

$=(x+y)\left(x^{2}-x y+y^{2}\right)$

(ii) It is known that,

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

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

$=(x-y)\left[(x-y)^{2}+3 x y\right]$

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

$=(x-y)\left(x^{2}+y^{2}+x y\right)$

$=(x-y)\left(x^{2}+x y+y^{2}\right)$