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