Question
Answer
$$(a-1)(a-2)(a-4)(a-7)(a-14)=a^5-28a^4+259a^3-988a^2+1540a-784$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(a-1)(a-2)(a-4)(a-7)(a-14)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1a^2-2a-a+2)(a-4)(a-7)(a-14) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1a^2-3a+2)(a-4)(a-7)(a-14) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(1a^3-4a^2-3a^2+12a+2a-8)(a-7)(a-14) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(1a^3-7a^2+14a-8)(a-7)(a-14) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(1a^4-14a^3+63a^2-106a+56)(a-14) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}a^5-28a^4+259a^3-988a^2+1540a-784\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{a-1}\right) $ by each term in $ \left( a-2\right) $. $$ \left( \color{blue}{a-1}\right) \cdot \left( a-2\right) = a^2-2a-a+2 $$ |
| ② | Combine like terms: $$ a^2 \color{blue}{-2a} \color{blue}{-a} +2 = a^2 \color{blue}{-3a} +2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{a^2-3a+2}\right) $ by each term in $ \left( a-4\right) $. $$ \left( \color{blue}{a^2-3a+2}\right) \cdot \left( a-4\right) = a^3-4a^2-3a^2+12a+2a-8 $$ |
| ④ | Combine like terms: $$ a^3 \color{blue}{-4a^2} \color{blue}{-3a^2} + \color{red}{12a} + \color{red}{2a} -8 = a^3 \color{blue}{-7a^2} + \color{red}{14a} -8 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{a^3-7a^2+14a-8}\right) $ by each term in $ \left( a-7\right) $. $$ \left( \color{blue}{a^3-7a^2+14a-8}\right) \cdot \left( a-7\right) = a^4-7a^3-7a^3+49a^2+14a^2-98a-8a+56 $$ |
| ⑥ | Combine like terms: $$ a^4 \color{blue}{-7a^3} \color{blue}{-7a^3} + \color{red}{49a^2} + \color{red}{14a^2} \color{green}{-98a} \color{green}{-8a} +56 = \\ = a^4 \color{blue}{-14a^3} + \color{red}{63a^2} \color{green}{-106a} +56 $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{a^4-14a^3+63a^2-106a+56}\right) $ by each term in $ \left( a-14\right) $. $$ \left( \color{blue}{a^4-14a^3+63a^2-106a+56}\right) \cdot \left( a-14\right) = \\ = a^5-14a^4-14a^4+196a^3+63a^3-882a^2-106a^2+1484a+56a-784 $$ |
| ⑧ | Combine like terms: $$ a^5 \color{blue}{-14a^4} \color{blue}{-14a^4} + \color{red}{196a^3} + \color{red}{63a^3} \color{green}{-882a^2} \color{green}{-106a^2} + \color{orange}{1484a} + \color{orange}{56a} -784 = \\ = a^5 \color{blue}{-28a^4} + \color{red}{259a^3} \color{green}{-988a^2} + \color{orange}{1540a} -784 $$ |
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