Question
Answer
$$(x-0)(x-20)(x-30)(x-40)(x-50)=x^5-140x^4+7100x^3-154000x^2+1200000x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-0)(x-20)(x-30)(x-40)(x-50)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-20x+0x+0)(x-30)(x-40)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-20x)(x-30)(x-40)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-30x^2-20x^2+600x)(x-40)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^3-50x^2+600x)(x-40)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(x^4-40x^3-50x^3+2000x^2+600x^2-24000x)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(x^4-90x^3+2600x^2-24000x)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}x^5-140x^4+7100x^3-154000x^2+1200000x\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x0}\right) $ by each term in $ \left( x-20\right) $. $$ \left( \color{blue}{x0}\right) \cdot \left( x-20\right) = x^2-20x0x0 $$ |
| ② | Combine like terms: $$ x^2 \color{blue}{-20x} \color{blue}{0x} 0 = x^2 \color{blue}{-20x} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-20x}\right) $ by each term in $ \left( x-30\right) $. $$ \left( \color{blue}{x^2-20x}\right) \cdot \left( x-30\right) = x^3-30x^2-20x^2+600x $$ |
| ④ | Combine like terms: $$ x^3 \color{blue}{-30x^2} \color{blue}{-20x^2} +600x = x^3 \color{blue}{-50x^2} +600x $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{x^3-50x^2+600x}\right) $ by each term in $ \left( x-40\right) $. $$ \left( \color{blue}{x^3-50x^2+600x}\right) \cdot \left( x-40\right) = x^4-40x^3-50x^3+2000x^2+600x^2-24000x $$ |
| ⑥ | Combine like terms: $$ x^4 \color{blue}{-40x^3} \color{blue}{-50x^3} + \color{red}{2000x^2} + \color{red}{600x^2} -24000x = x^4 \color{blue}{-90x^3} + \color{red}{2600x^2} -24000x $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{x^4-90x^3+2600x^2-24000x}\right) $ by each term in $ \left( x-50\right) $. $$ \left( \color{blue}{x^4-90x^3+2600x^2-24000x}\right) \cdot \left( x-50\right) = \\ = x^5-50x^4-90x^4+4500x^3+2600x^3-130000x^2-24000x^2+1200000x $$ |
| ⑧ | Combine like terms: $$ x^5 \color{blue}{-50x^4} \color{blue}{-90x^4} + \color{red}{4500x^3} + \color{red}{2600x^3} \color{green}{-130000x^2} \color{green}{-24000x^2} +1200000x = \\ = x^5 \color{blue}{-140x^4} + \color{red}{7100x^3} \color{green}{-154000x^2} +1200000x $$ |
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