Question
Answer
$$(1+2x+3x^2)(4+5x+2x^2)(2+2x+2x^2)(2+3x+3x^2)=36x^8+186x^7+468x^6+730x^5+778x^4+572x^3+290x^2+92x+16$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+2x+3x^2)(4+5x+2x^2)(2+2x+2x^2)(2+3x+3x^2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(6x^4+19x^3+24x^2+13x+4)(2+2x+2x^2)(2+3x+3x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(12x^6+50x^5+98x^4+112x^3+82x^2+34x+8)(2+3x+3x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}36x^8+186x^7+468x^6+730x^5+778x^4+572x^3+290x^2+92x+16\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1+2x+3x^2}\right) $ by each term in $ \left( 4+5x+2x^2\right) $. $$ \left( \color{blue}{1+2x+3x^2}\right) \cdot \left( 4+5x+2x^2\right) = 4+5x+2x^2+8x+10x^2+4x^3+12x^2+15x^3+6x^4 $$ |
| ② | Combine like terms: $$ 4+ \color{blue}{5x} + \color{red}{2x^2} + \color{blue}{8x} + \color{green}{10x^2} + \color{orange}{4x^3} + \color{green}{12x^2} + \color{orange}{15x^3} +6x^4 = \\ = 6x^4+ \color{orange}{19x^3} + \color{green}{24x^2} + \color{blue}{13x} +4 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{6x^4+19x^3+24x^2+13x+4}\right) $ by each term in $ \left( 2+2x+2x^2\right) $. $$ \left( \color{blue}{6x^4+19x^3+24x^2+13x+4}\right) \cdot \left( 2+2x+2x^2\right) = \\ = 12x^4+12x^5+12x^6+38x^3+38x^4+38x^5+48x^2+48x^3+48x^4+26x+26x^2+26x^3+8+8x+8x^2 $$ |
| ④ | Combine like terms: $$ \color{blue}{12x^4} + \color{red}{12x^5} +12x^6+ \color{green}{38x^3} + \color{orange}{38x^4} + \color{red}{38x^5} + \color{blue}{48x^2} + \color{red}{48x^3} + \color{orange}{48x^4} + \color{green}{26x} + \color{orange}{26x^2} + \color{red}{26x^3} +8+ \color{green}{8x} + \color{orange}{8x^2} = \\ = 12x^6+ \color{red}{50x^5} + \color{orange}{98x^4} + \color{red}{112x^3} + \color{orange}{82x^2} + \color{green}{34x} +8 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{12x^6+50x^5+98x^4+112x^3+82x^2+34x+8}\right) $ by each term in $ \left( 2+3x+3x^2\right) $. $$ \left( \color{blue}{12x^6+50x^5+98x^4+112x^3+82x^2+34x+8}\right) \cdot \left( 2+3x+3x^2\right) = \\ = 24x^6+36x^7+36x^8+100x^5+150x^6+150x^7+196x^4+294x^5+294x^6+224x^3+336x^4+336x^5+164x^2+246x^3+246x^4+68x+102x^2+102x^3+16+24x+24x^2 $$ |
| ⑥ | Combine like terms: $$ \color{blue}{24x^6} + \color{red}{36x^7} +36x^8+ \color{green}{100x^5} + \color{orange}{150x^6} + \color{red}{150x^7} + \color{blue}{196x^4} + \color{red}{294x^5} + \color{orange}{294x^6} + \color{green}{224x^3} + \color{orange}{336x^4} + \color{red}{336x^5} + \color{blue}{164x^2} + \color{red}{246x^3} + \color{orange}{246x^4} + \color{green}{68x} + \color{orange}{102x^2} + \color{red}{102x^3} +16+ \color{green}{24x} + \color{orange}{24x^2} = \\ = 36x^8+ \color{red}{186x^7} + \color{orange}{468x^6} + \color{red}{730x^5} + \color{orange}{778x^4} + \color{red}{572x^3} + \color{orange}{290x^2} + \color{green}{92x} +16 $$ |
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