Question
Answer
$$(1+16x+72x^2+96x^3+24x^4)^2=576x^8+4608x^7+12672x^6+14592x^5+8304x^4+2496x^3+400x^2+32x+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+16x+72x^2+96x^3+24x^4)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}576x^8+4608x^7+12672x^6+14592x^5+8304x^4+2496x^3+400x^2+32x+1\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1+16x+72x^2+96x^3+24x^4}\right) $ by each term in $ \left( 1+16x+72x^2+96x^3+24x^4\right) $. $$ \left( \color{blue}{1+16x+72x^2+96x^3+24x^4}\right) \cdot \left( 1+16x+72x^2+96x^3+24x^4\right) = \\ = 1+16x+72x^2+96x^3+24x^4+16x+256x^2+1152x^3+1536x^4+384x^5+72x^2+1152x^3+5184x^4+6912x^5+1728x^6+96x^3+1536x^4+6912x^5+9216x^6+2304x^7+24x^4+384x^5+1728x^6+2304x^7+576x^8 $$ |
| ② | Combine like terms: $$ 1+ \color{blue}{16x} + \color{red}{72x^2} + \color{green}{96x^3} + \color{orange}{24x^4} + \color{blue}{16x} + \color{blue}{256x^2} + \color{red}{1152x^3} + \color{green}{1536x^4} + \color{orange}{384x^5} + \color{blue}{72x^2} + \color{blue}{1152x^3} + \color{red}{5184x^4} + \color{green}{6912x^5} + \color{orange}{1728x^6} + \color{blue}{96x^3} + \color{blue}{1536x^4} + \color{red}{6912x^5} + \color{green}{9216x^6} + \color{orange}{2304x^7} + \color{blue}{24x^4} + \color{red}{384x^5} + \color{green}{1728x^6} + \color{orange}{2304x^7} +576x^8 = \\ = 576x^8+ \color{orange}{4608x^7} + \color{green}{12672x^6} + \color{red}{14592x^5} + \color{blue}{8304x^4} + \color{blue}{2496x^3} + \color{blue}{400x^2} + \color{blue}{32x} +1 $$ |
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