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

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The identity $$ x^{3} + y^{3} = (x+y)(x^{2} - xy + y^{2}) $$ is verified by expanding and simplifying the right-hand side to show it equals the left-hand side. To verify the identity $$ x^{3} + y^{3} = (x+y)(x^{2} - xy + y^{2}), $$ we can start by expanding the right-hand side. 1. **Expand the right-hand side**: $$ (x+y)(x^{2} - xy + y^{2}) = x(x^{2} - xy + y^{2}) + y(x^{2} - xy + y^{2}). $$ Now, distribute $$x$$ and $$y$$: $$ = x \cdot x^{2} - x \cdot xy + x \cdot y^{2} + y \cdot x^{2} - y \cdot xy + y \cdot y^{2}. $$ This simplifies to: $$ = x^{3} - x^{2}y + xy^{2} + yx^{2} - y^{2}x + y^{3}. $$ Notice that $$-x^{2}y + xy^{2} + yx^{2} - y^{2}x$$ can be rearranged: $$ = x^{3} + y^{3} + (yx^{2} - x^{2}y) + (xy^{2} - y^{2}x). $$ The terms $$yx^{2} - x^{2}y$$ and $$xy^{2} - y^{2}x$$ both equal zero, since they are just rearrangements of the same terms. Thus, we have: $$ = x^{3} + y^{3}. $$ 2. **Conclusion**: Since we have shown that $$ (x+y)(x^{2} - xy + y^{2}) = x^{3} + y^{3}, $$ the identity $$ x^{3} + y^{3} = (x+y)(x^{2} - xy + y^{2}) $$ is verified.

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

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

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