Question
Answer
$$k(k-1)^2(k-3)^2(k-3)(k-4^2)=k^7-27k^6+222k^5-826k^4+1521k^3-1323k^2+432k$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}k(k-1)^2(k-3)^2(k-3)(k-4^2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}k(1k^2-2k+1)(1k^2-6k+9)(k-3)(k-16) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1k^3-2k^2+k)(1k^2-6k+9)(k-3)(k-16) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(1k^5-8k^4+22k^3-24k^2+9k)(k-3)(k-16) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(1k^6-11k^5+46k^4-90k^3+81k^2-27k)(k-16) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}k^7-27k^6+222k^5-826k^4+1521k^3-1323k^2+432k\end{aligned} $$ | |
| ① | Find $ \left(k-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}{ k } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(k-1\right)^2 = \color{blue}{k^2} -2 \cdot k \cdot 1 + \color{red}{1^2} = k^2-2k+1\end{aligned} $$Find $ \left(k-3\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ k } $ and $ B = \color{red}{ 3 }$. $$ \begin{aligned}\left(k-3\right)^2 = \color{blue}{k^2} -2 \cdot k \cdot 3 + \color{red}{3^2} = k^2-6k+9\end{aligned} $$$$ k^2 = 1^2k^2 = k^2 $$ |
| ② | Multiply $ \color{blue}{k} $ by $ \left( k^2-2k+1\right) $ $$ \color{blue}{k} \cdot \left( k^2-2k+1\right) = k^3-2k^2+k $$ |
| ③ | Multiply each term of $ \left( \color{blue}{k^3-2k^2+k}\right) $ by each term in $ \left( k^2-6k+9\right) $. $$ \left( \color{blue}{k^3-2k^2+k}\right) \cdot \left( k^2-6k+9\right) = k^5-6k^4+9k^3-2k^4+12k^3-18k^2+k^3-6k^2+9k $$ |
| ④ | Combine like terms: $$ k^5 \color{blue}{-6k^4} + \color{red}{9k^3} \color{blue}{-2k^4} + \color{green}{12k^3} \color{orange}{-18k^2} + \color{green}{k^3} \color{orange}{-6k^2} +9k = \\ = k^5 \color{blue}{-8k^4} + \color{green}{22k^3} \color{orange}{-24k^2} +9k $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{k^5-8k^4+22k^3-24k^2+9k}\right) $ by each term in $ \left( k-3\right) $. $$ \left( \color{blue}{k^5-8k^4+22k^3-24k^2+9k}\right) \cdot \left( k-3\right) = \\ = k^6-3k^5-8k^5+24k^4+22k^4-66k^3-24k^3+72k^2+9k^2-27k $$ |
| ⑥ | Combine like terms: $$ k^6 \color{blue}{-3k^5} \color{blue}{-8k^5} + \color{red}{24k^4} + \color{red}{22k^4} \color{green}{-66k^3} \color{green}{-24k^3} + \color{orange}{72k^2} + \color{orange}{9k^2} -27k = \\ = k^6 \color{blue}{-11k^5} + \color{red}{46k^4} \color{green}{-90k^3} + \color{orange}{81k^2} -27k $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{k^6-11k^5+46k^4-90k^3+81k^2-27k}\right) $ by each term in $ \left( k-16\right) $. $$ \left( \color{blue}{k^6-11k^5+46k^4-90k^3+81k^2-27k}\right) \cdot \left( k-16\right) = \\ = k^7-16k^6-11k^6+176k^5+46k^5-736k^4-90k^4+1440k^3+81k^3-1296k^2-27k^2+432k $$ |
| ⑧ | Combine like terms: $$ k^7 \color{blue}{-16k^6} \color{blue}{-11k^6} + \color{red}{176k^5} + \color{red}{46k^5} \color{green}{-736k^4} \color{green}{-90k^4} + \color{orange}{1440k^3} + \color{orange}{81k^3} \color{blue}{-1296k^2} \color{blue}{-27k^2} +432k = \\ = k^7 \color{blue}{-27k^6} + \color{red}{222k^5} \color{green}{-826k^4} + \color{orange}{1521k^3} \color{blue}{-1323k^2} +432k $$ |
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