Question
Answer
$$(a+b+c)(a^2+b^2+c^2-ab-ac-bc)=a^3-3abc+b^3+c^3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(a+b+c)(a^2+b^2+c^2-ab-ac-bc)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}a^3-3abc+b^3+c^3\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{a+b+c}\right) $ by each term in $ \left( a^2+b^2+c^2-ab-ac-bc\right) $. $$ \left( \color{blue}{a+b+c}\right) \cdot \left( a^2+b^2+c^2-ab-ac-bc\right) = \\ = a^3+ \cancel{ab^2}+ \cancel{ac^2} -\cancel{a^2b} -\cancel{a^2c}-abc+ \cancel{a^2b}+b^3+ \cancel{bc^2} -\cancel{ab^2}-abc -\cancel{b^2c}+ \cancel{a^2c}+ \cancel{b^2c}+c^3-abc -\cancel{ac^2} -\cancel{bc^2} $$ |
| ② | Combine like terms: $$ a^3+ \, \color{blue}{ \cancel{ab^2}} \,+ \, \color{green}{ \cancel{ac^2}} \, \, \color{blue}{ -\cancel{a^2b}} \, \, \color{green}{ -\cancel{a^2c}} \, \color{blue}{-abc} + \, \color{blue}{ \cancel{a^2b}} \,+b^3+ \, \color{red}{ \cancel{bc^2}} \, \, \color{blue}{ -\cancel{ab^2}} \, \color{orange}{-abc} \, \color{blue}{ -\cancel{b^2c}} \,+ \, \color{green}{ \cancel{a^2c}} \,+ \, \color{blue}{ \cancel{b^2c}} \,+c^3 \color{orange}{-abc} \, \color{green}{ -\cancel{ac^2}} \, \, \color{red}{ -\cancel{bc^2}} \, = a^3 \color{orange}{-3abc} +b^3+c^3 $$ |
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