Question
Answer
$$(a+b+c)^3=a^3+3a^2b+3a^2c+3ab^2+6abc+3ac^2+b^3+3b^2c+3bc^2+c^3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(a+b+c)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}a^3+3a^2b+3a^2c+3ab^2+6abc+3ac^2+b^3+3b^2c+3bc^2+c^3\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{a+b+c}\right) $ by each term in $ \left( a+b+c\right) $. $$ \left( \color{blue}{a+b+c}\right) \cdot \left( a+b+c\right) = a^2+ab+ac+ab+b^2+bc+ac+bc+c^2 $$ |
| ② | Combine like terms: $$ a^2+ \color{blue}{ab} + \color{red}{ac} + \color{blue}{ab} +b^2+ \color{green}{bc} + \color{red}{ac} + \color{green}{bc} +c^2 = \\ = a^2+ \color{blue}{2ab} + \color{red}{2ac} +b^2+ \color{green}{2bc} +c^2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{a^2+2ab+2ac+b^2+2bc+c^2}\right) $ by each term in $ \left( a+b+c\right) $. $$ \left( \color{blue}{a^2+2ab+2ac+b^2+2bc+c^2}\right) \cdot \left( a+b+c\right) = \\ = a^3+a^2b+a^2c+2a^2b+2ab^2+2abc+2a^2c+2abc+2ac^2+ab^2+b^3+b^2c+2abc+2b^2c+2bc^2+ac^2+bc^2+c^3 $$ |
| ④ | Combine like terms: $$ a^3+ \color{blue}{a^2b} + \color{red}{a^2c} + \color{blue}{2a^2b} + \color{green}{2ab^2} + \color{orange}{2abc} + \color{red}{2a^2c} + \color{blue}{2abc} + \color{red}{2ac^2} + \color{green}{ab^2} +b^3+ \color{green}{b^2c} + \color{blue}{2abc} + \color{green}{2b^2c} + \color{orange}{2bc^2} + \color{red}{ac^2} + \color{orange}{bc^2} +c^3 = \\ = a^3+ \color{blue}{3a^2b} + \color{red}{3a^2c} + \color{green}{3ab^2} + \color{blue}{6abc} + \color{red}{3ac^2} +b^3+ \color{green}{3b^2c} + \color{orange}{3bc^2} +c^3 $$ |
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