Observations are important: Pre RMO and RMO : algebra

Posted by Nalin Pithwa

We know the following facts very well:

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

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

But, you can quickly verify that:

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

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

Whereas:

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

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

I call it — simply stunning beauty of elementary algebra of factorizations and expansions

More later,

Nalin Pithwa

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