Question
Answer
$$(1+8u+16u^2+16u^3)(1+4u)^3=1024u^6+1792u^5+1472u^4+656u^3+160u^2+20u+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+8u+16u^2+16u^3)(1+4u)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1+8u+16u^2+16u^3)(1+12u+48u^2+64u^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}1024u^6+1792u^5+1472u^4+656u^3+160u^2+20u+1\end{aligned} $$ | |
| ① | Find $ \left(1+4u\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 1 $ and $ B = 4u $. $$ \left(1+4u\right)^3 = 1^3+3 \cdot 1^2 \cdot 4u + 3 \cdot 1 \cdot \left( 4u \right)^2+\left( 4u \right)^3 = 1+12u+48u^2+64u^3 $$ |
| ② | Multiply each term of $ \left( \color{blue}{1+8u+16u^2+16u^3}\right) $ by each term in $ \left( 1+12u+48u^2+64u^3\right) $. $$ \left( \color{blue}{1+8u+16u^2+16u^3}\right) \cdot \left( 1+12u+48u^2+64u^3\right) = \\ = 1+12u+48u^2+64u^3+8u+96u^2+384u^3+512u^4+16u^2+192u^3+768u^4+1024u^5+16u^3+192u^4+768u^5+1024u^6 $$ |
| ③ | Combine like terms: $$ 1+ \color{blue}{12u} + \color{red}{48u^2} + \color{green}{64u^3} + \color{blue}{8u} + \color{orange}{96u^2} + \color{blue}{384u^3} + \color{red}{512u^4} + \color{orange}{16u^2} + \color{green}{192u^3} + \color{orange}{768u^4} + \color{blue}{1024u^5} + \color{green}{16u^3} + \color{orange}{192u^4} + \color{blue}{768u^5} +1024u^6 = \\ = 1024u^6+ \color{blue}{1792u^5} + \color{orange}{1472u^4} + \color{green}{656u^3} + \color{orange}{160u^2} + \color{blue}{20u} +1 $$ |
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