Question
Answer
$$(x-49)(x-55)(x-60.5)(x-70.2)(x-78)=x^5-312x^4+38667x^3-2379520x^2+72716700x-882882000$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-49)(x-55)(x-60.5)(x-70.2)(x-78)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-55x-49x+2695)(x-60.5)(x-70.2)(x-78) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-104x+2695)(x-60.5)(x-70.2)(x-78) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-60x^2-104x^2+6240x+2695x-161700)(x-70.2)(x-78) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^3-164x^2+8935x-161700)(x-70.2)(x-78) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(x^4-234x^3+20415x^2-787150x+11319000)(x-78) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}x^5-312x^4+38667x^3-2379520x^2+72716700x-882882000\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-49}\right) $ by each term in $ \left( x-55\right) $. $$ \left( \color{blue}{x-49}\right) \cdot \left( x-55\right) = x^2-55x-49x+2695 $$ |
| ② | Combine like terms: $$ x^2 \color{blue}{-55x} \color{blue}{-49x} +2695 = x^2 \color{blue}{-104x} +2695 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-104x+2695}\right) $ by each term in $ \left( x-60\right) $. $$ \left( \color{blue}{x^2-104x+2695}\right) \cdot \left( x-60\right) = x^3-60x^2-104x^2+6240x+2695x-161700 $$ |
| ④ | Combine like terms: $$ x^3 \color{blue}{-60x^2} \color{blue}{-104x^2} + \color{red}{6240x} + \color{red}{2695x} -161700 = x^3 \color{blue}{-164x^2} + \color{red}{8935x} -161700 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{x^3-164x^2+8935x-161700}\right) $ by each term in $ \left( x-70\right) $. $$ \left( \color{blue}{x^3-164x^2+8935x-161700}\right) \cdot \left( x-70\right) = \\ = x^4-70x^3-164x^3+11480x^2+8935x^2-625450x-161700x+11319000 $$ |
| ⑥ | Combine like terms: $$ x^4 \color{blue}{-70x^3} \color{blue}{-164x^3} + \color{red}{11480x^2} + \color{red}{8935x^2} \color{green}{-625450x} \color{green}{-161700x} +11319000 = \\ = x^4 \color{blue}{-234x^3} + \color{red}{20415x^2} \color{green}{-787150x} +11319000 $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{x^4-234x^3+20415x^2-787150x+11319000}\right) $ by each term in $ \left( x-78\right) $. $$ \left( \color{blue}{x^4-234x^3+20415x^2-787150x+11319000}\right) \cdot \left( x-78\right) = \\ = x^5-78x^4-234x^4+18252x^3+20415x^3-1592370x^2-787150x^2+61397700x+11319000x-882882000 $$ |
| ⑧ | Combine like terms: $$ x^5 \color{blue}{-78x^4} \color{blue}{-234x^4} + \color{red}{18252x^3} + \color{red}{20415x^3} \color{green}{-1592370x^2} \color{green}{-787150x^2} + \color{orange}{61397700x} + \color{orange}{11319000x} -882882000 = \\ = x^5 \color{blue}{-312x^4} + \color{red}{38667x^3} \color{green}{-2379520x^2} + \color{orange}{72716700x} -882882000 $$ |
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