Question
Answer
$$(x+1)^4+(x-1)^4=2x^4+12x^2+2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+1)^4+(x-1)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}x^4+4x^3+6x^2+4x+1+x^4-4x^3+6x^2-4x+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}2x^4+12x^2+2\end{aligned} $$ | |
| ① | $$ (x+1)^4 = (x+1)^2 \cdot (x+1)^2 $$ |
| ② | Find $ \left(x+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}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2+2x+1}\right) $ by each term in $ \left( x^2+2x+1\right) $. $$ \left( \color{blue}{x^2+2x+1}\right) \cdot \left( x^2+2x+1\right) = x^4+2x^3+x^2+2x^3+4x^2+2x+x^2+2x+1 $$ |
| ④ | Combine like terms: $$ x^4+ \color{blue}{2x^3} + \color{red}{x^2} + \color{blue}{2x^3} + \color{green}{4x^2} + \color{orange}{2x} + \color{green}{x^2} + \color{orange}{2x} +1 = x^4+ \color{blue}{4x^3} + \color{green}{6x^2} + \color{orange}{4x} +1 $$$$ (x-1)^4 = (x-1)^2 \cdot (x-1)^2 $$ |
| ⑤ | Find $ \left(x-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}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x-1\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 1 + \color{red}{1^2} = x^2-2x+1\end{aligned} $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{x^2-2x+1}\right) $ by each term in $ \left( x^2-2x+1\right) $. $$ \left( \color{blue}{x^2-2x+1}\right) \cdot \left( x^2-2x+1\right) = x^4-2x^3+x^2-2x^3+4x^2-2x+x^2-2x+1 $$ |
| ⑦ | Combine like terms: $$ x^4 \color{blue}{-2x^3} + \color{red}{x^2} \color{blue}{-2x^3} + \color{green}{4x^2} \color{orange}{-2x} + \color{green}{x^2} \color{orange}{-2x} +1 = x^4 \color{blue}{-4x^3} + \color{green}{6x^2} \color{orange}{-4x} +1 $$ |
| ⑧ | Combine like terms: $$ \color{blue}{x^4} + \, \color{red}{ \cancel{4x^3}} \,+ \color{orange}{6x^2} + \, \color{blue}{ \cancel{4x}} \,+ \color{green}{1} + \color{blue}{x^4} \, \color{red}{ -\cancel{4x^3}} \,+ \color{orange}{6x^2} \, \color{blue}{ -\cancel{4x}} \,+ \color{green}{1} = \color{blue}{2x^4} + \color{orange}{12x^2} + \color{green}{2} $$ |
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