Question

Verify the relation, $x^3+y^3=(x+y)(x^2-xy+y^2)$ using some non-zero positive integers and check by actual multiplication. Can you call this equation an identity?

Answer 1

To prove: $x^3+y^3=(x+y)(x^2-xy+y^2)$

Consider the right hand side ($RHS$) and expand it as follows:

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

Hence, the equality of the given relation is verified.

Yes, we can call it an identity, as the equality will hold for all values of $x$ and $y$.

For example:

Let us take $x=2$ and $y=1$ and substitute in the $LHS$ and $RHS$ of the equation to prove the relation:

$LHS=x^3+y^3=2^3+1^3=9$ and

$RHS=(2+1)(2^2-(2\times 1)+1^2)=3(5-2)=3\times 3=9$

Therefore, $LHS=RHS$

Hence, the equality holds and $x^3+y^3=(x+y)(x^2-xy+y^2)$ can be used as an identity.