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