Question
Verify : $x^3+y^3=(x+y)\left(x^2-x y+y^2\right)$
✓
Answer
We know that
$(x+y)^3=x^3+y^3+3 x y(x+y)\left\{\text { Using Identity }(a+b)^3=a^3+b^3+3 a b(a+b)\right\}$
$\Rightarrow x^3+y^3=(x+y)^3-3 x y(x+y)$
$\Rightarrow x^3+y^3=(x+y)\left\{(x+y)^2-3 x y\right\}$
$\Rightarrow x^3+y^3=(x+y)\left(x^2+2 x y+y^2-3 x y\right)\left\{\text { Using Identity }(a+b)^2=a^2+2 a b+b^2\right\}$
$\Rightarrow x^3+y^3=(x+y)\left(x^2-x y+y^2\right)$
$(x+y)^3=x^3+y^3+3 x y(x+y)\left\{\text { Using Identity }(a+b)^3=a^3+b^3+3 a b(a+b)\right\}$
$\Rightarrow x^3+y^3=(x+y)^3-3 x y(x+y)$
$\Rightarrow x^3+y^3=(x+y)\left\{(x+y)^2-3 x y\right\}$
$\Rightarrow x^3+y^3=(x+y)\left(x^2+2 x y+y^2-3 x y\right)\left\{\text { Using Identity }(a+b)^2=a^2+2 a b+b^2\right\}$
$\Rightarrow x^3+y^3=(x+y)\left(x^2-x y+y^2\right)$
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