Answer
$$(x+y+z)(x^2+y^2+z^2-xy-yz-xz)=x^3-3xyz+y^3+z^3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+y+z)(x^2+y^2+z^2-xy-yz-xz)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^3-3xyz+y^3+z^3\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x+y+z}\right) $ by each term in $ \left( x^2+y^2+z^2-xy-yz-xz\right) $. $$ \left( \color{blue}{x+y+z}\right) \cdot \left( x^2+y^2+z^2-xy-yz-xz\right) = \\ = x^3+ \cancel{xy^2}+ \cancel{xz^2} -\cancel{x^2y}-xyz -\cancel{x^2z}+ \cancel{x^2y}+y^3+ \cancel{yz^2} -\cancel{xy^2} -\cancel{y^2z}-xyz+ \cancel{x^2z}+ \cancel{y^2z}+z^3-xyz -\cancel{yz^2} -\cancel{xz^2} $$ |
| ② | Combine like terms: $$ x^3+ \, \color{blue}{ \cancel{xy^2}} \,+ \, \color{green}{ \cancel{xz^2}} \, \, \color{blue}{ -\cancel{x^2y}} \, \color{green}{-xyz} \, \color{orange}{ -\cancel{x^2z}} \,+ \, \color{blue}{ \cancel{x^2y}} \,+y^3+ \, \color{red}{ \cancel{yz^2}} \, \, \color{blue}{ -\cancel{xy^2}} \, \, \color{orange}{ -\cancel{y^2z}} \, \color{red}{-xyz} + \, \color{orange}{ \cancel{x^2z}} \,+ \, \color{orange}{ \cancel{y^2z}} \,+z^3 \color{red}{-xyz} \, \color{red}{ -\cancel{yz^2}} \, \, \color{green}{ -\cancel{xz^2}} \, = x^3 \color{red}{-3xyz} +y^3+z^3 $$ |
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