Question
Answer
$$(n-3)(n-2)(n-1)n(n+1)=n^5-5n^4+5n^3+5n^2-6n$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(n-3)(n-2)(n-1)n(n+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1n^2-2n-3n+6)(n-1)n(n+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1n^2-5n+6)(n-1)n(n+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(1n^3-n^2-5n^2+5n+6n-6)n(n+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(1n^3-6n^2+11n-6)n(n+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(1n^4-6n^3+11n^2-6n)(n+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}n^5-5n^4+5n^3+5n^2-6n\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{n-3}\right) $ by each term in $ \left( n-2\right) $. $$ \left( \color{blue}{n-3}\right) \cdot \left( n-2\right) = n^2-2n-3n+6 $$ |
| ② | Combine like terms: $$ n^2 \color{blue}{-2n} \color{blue}{-3n} +6 = n^2 \color{blue}{-5n} +6 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{n^2-5n+6}\right) $ by each term in $ \left( n-1\right) $. $$ \left( \color{blue}{n^2-5n+6}\right) \cdot \left( n-1\right) = n^3-n^2-5n^2+5n+6n-6 $$ |
| ④ | Combine like terms: $$ n^3 \color{blue}{-n^2} \color{blue}{-5n^2} + \color{red}{5n} + \color{red}{6n} -6 = n^3 \color{blue}{-6n^2} + \color{red}{11n} -6 $$ |
| ⑤ | $$ \left( \color{blue}{n^3-6n^2+11n-6}\right) \cdot n = n^4-6n^3+11n^2-6n $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{n^4-6n^3+11n^2-6n}\right) $ by each term in $ \left( n+1\right) $. $$ \left( \color{blue}{n^4-6n^3+11n^2-6n}\right) \cdot \left( n+1\right) = n^5+n^4-6n^4-6n^3+11n^3+11n^2-6n^2-6n $$ |
| ⑦ | Combine like terms: $$ n^5+ \color{blue}{n^4} \color{blue}{-6n^4} \color{red}{-6n^3} + \color{red}{11n^3} + \color{green}{11n^2} \color{green}{-6n^2} -6n = n^5 \color{blue}{-5n^4} + \color{red}{5n^3} + \color{green}{5n^2} -6n $$ |
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