Question
Answer
$$(x-0)(x-10)(x-20)(x-30)(x-50)=x^5-110x^4+4100x^3-61000x^2+300000x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-0)(x-10)(x-20)(x-30)(x-50)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-10x+0x+0)(x-20)(x-30)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-10x)(x-20)(x-30)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-20x^2-10x^2+200x)(x-30)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^3-30x^2+200x)(x-30)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(x^4-30x^3-30x^3+900x^2+200x^2-6000x)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(x^4-60x^3+1100x^2-6000x)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}x^5-110x^4+4100x^3-61000x^2+300000x\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x0}\right) $ by each term in $ \left( x-10\right) $. $$ \left( \color{blue}{x0}\right) \cdot \left( x-10\right) = x^2-10x0x0 $$ |
| ② | Combine like terms: $$ x^2 \color{blue}{-10x} \color{blue}{0x} 0 = x^2 \color{blue}{-10x} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-10x}\right) $ by each term in $ \left( x-20\right) $. $$ \left( \color{blue}{x^2-10x}\right) \cdot \left( x-20\right) = x^3-20x^2-10x^2+200x $$ |
| ④ | Combine like terms: $$ x^3 \color{blue}{-20x^2} \color{blue}{-10x^2} +200x = x^3 \color{blue}{-30x^2} +200x $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{x^3-30x^2+200x}\right) $ by each term in $ \left( x-30\right) $. $$ \left( \color{blue}{x^3-30x^2+200x}\right) \cdot \left( x-30\right) = x^4-30x^3-30x^3+900x^2+200x^2-6000x $$ |
| ⑥ | Combine like terms: $$ x^4 \color{blue}{-30x^3} \color{blue}{-30x^3} + \color{red}{900x^2} + \color{red}{200x^2} -6000x = x^4 \color{blue}{-60x^3} + \color{red}{1100x^2} -6000x $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{x^4-60x^3+1100x^2-6000x}\right) $ by each term in $ \left( x-50\right) $. $$ \left( \color{blue}{x^4-60x^3+1100x^2-6000x}\right) \cdot \left( x-50\right) = \\ = x^5-50x^4-60x^4+3000x^3+1100x^3-55000x^2-6000x^2+300000x $$ |
| ⑧ | Combine like terms: $$ x^5 \color{blue}{-50x^4} \color{blue}{-60x^4} + \color{red}{3000x^3} + \color{red}{1100x^3} \color{green}{-55000x^2} \color{green}{-6000x^2} +300000x = \\ = x^5 \color{blue}{-110x^4} + \color{red}{4100x^3} \color{green}{-61000x^2} +300000x $$ |
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