Question
Answer
$$k(k-1)(k-2)(k-3)+k(k-1)(k-2)=k^4-5k^3+8k^2-4k$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}k(k-1)(k-2)(k-3)+k(k-1)(k-2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1k^2-k)(k-2)(k-3)+(1k^2-k)(k-2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1k^3-2k^2-k^2+2k)(k-3)+k^3-2k^2-k^2+2k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(1k^3-3k^2+2k)(k-3)+k^3-3k^2+2k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}k^4-3k^3-3k^3+9k^2+2k^2-6k+k^3-3k^2+2k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}k^4-6k^3+11k^2-6k+k^3-3k^2+2k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}k^4-5k^3+8k^2-4k\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{k} $ by $ \left( k-1\right) $ $$ \color{blue}{k} \cdot \left( k-1\right) = k^2-k $$Multiply $ \color{blue}{k} $ by $ \left( k-1\right) $ $$ \color{blue}{k} \cdot \left( k-1\right) = k^2-k $$ |
| ② | Multiply each term of $ \left( \color{blue}{k^2-k}\right) $ by each term in $ \left( k-2\right) $. $$ \left( \color{blue}{k^2-k}\right) \cdot \left( k-2\right) = k^3-2k^2-k^2+2k $$Multiply each term of $ \left( \color{blue}{k^2-k}\right) $ by each term in $ \left( k-2\right) $. $$ \left( \color{blue}{k^2-k}\right) \cdot \left( k-2\right) = k^3-2k^2-k^2+2k $$ |
| ③ | Combine like terms: $$ k^3 \color{blue}{-2k^2} \color{blue}{-k^2} +2k = k^3 \color{blue}{-3k^2} +2k $$Combine like terms: $$ k^3 \color{blue}{-2k^2} \color{blue}{-k^2} +2k = k^3 \color{blue}{-3k^2} +2k $$ |
| ④ | Multiply each term of $ \left( \color{blue}{k^3-3k^2+2k}\right) $ by each term in $ \left( k-3\right) $. $$ \left( \color{blue}{k^3-3k^2+2k}\right) \cdot \left( k-3\right) = k^4-3k^3-3k^3+9k^2+2k^2-6k $$ |
| ⑤ | Combine like terms: $$ k^4 \color{blue}{-3k^3} \color{blue}{-3k^3} + \color{red}{9k^2} + \color{red}{2k^2} -6k = k^4 \color{blue}{-6k^3} + \color{red}{11k^2} -6k $$ |
| ⑥ | Combine like terms: $$ k^4 \color{blue}{-6k^3} + \color{red}{11k^2} \color{green}{-6k} + \color{blue}{k^3} \color{red}{-3k^2} + \color{green}{2k} = k^4 \color{blue}{-5k^3} + \color{red}{8k^2} \color{green}{-4k} $$ |
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