Question
Answer
$$(1+x^2+2a)^3=x^6+6ax^4+12a^2x^2+3x^4+8a^3+12ax^2+12a^2+3x^2+6a+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+x^2+2a)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^6+6ax^4+12a^2x^2+3x^4+8a^3+12ax^2+12a^2+3x^2+6a+1\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1+x^2+2a}\right) $ by each term in $ \left( 1+x^2+2a\right) $. $$ \left( \color{blue}{1+x^2+2a}\right) \cdot \left( 1+x^2+2a\right) = 1+x^2+2a+x^2+x^4+2ax^2+2a+2ax^2+4a^2 $$ |
| ② | Combine like terms: $$ 1+ \color{blue}{x^2} + \color{red}{2a} + \color{blue}{x^2} +x^4+ \color{green}{2ax^2} + \color{red}{2a} + \color{green}{2ax^2} +4a^2 = \\ = x^4+ \color{green}{4ax^2} +4a^2+ \color{blue}{2x^2} + \color{red}{4a} +1 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^4+4ax^2+4a^2+2x^2+4a+1}\right) $ by each term in $ \left( 1+x^2+2a\right) $. $$ \left( \color{blue}{x^4+4ax^2+4a^2+2x^2+4a+1}\right) \cdot \left( 1+x^2+2a\right) = \\ = x^4+x^6+2ax^4+4ax^2+4ax^4+8a^2x^2+4a^2+4a^2x^2+8a^3+2x^2+2x^4+4ax^2+4a+4ax^2+8a^2+1+x^2+2a $$ |
| ④ | Combine like terms: $$ \color{blue}{x^4} +x^6+ \color{red}{2ax^4} + \color{green}{4ax^2} + \color{red}{4ax^4} + \color{orange}{8a^2x^2} + \color{blue}{4a^2} + \color{orange}{4a^2x^2} +8a^3+ \color{red}{2x^2} + \color{blue}{2x^4} + \color{green}{4ax^2} + \color{orange}{4a} + \color{green}{4ax^2} + \color{blue}{8a^2} +1+ \color{red}{x^2} + \color{orange}{2a} = \\ = x^6+ \color{red}{6ax^4} + \color{orange}{12a^2x^2} + \color{blue}{3x^4} +8a^3+ \color{green}{12ax^2} + \color{blue}{12a^2} + \color{red}{3x^2} + \color{orange}{6a} +1 $$ |
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