Question
Answer
$$(x-10)(x-20)(x-30)(x-40)(x-50)=x^5-150x^4+8500x^3-225000x^2+2740000x-12000000$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-10)(x-20)(x-30)(x-40)(x-50)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-20x-10x+200)(x-30)(x-40)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-30x+200)(x-30)(x-40)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-30x^2-30x^2+900x+200x-6000)(x-40)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^3-60x^2+1100x-6000)(x-40)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(x^4-100x^3+3500x^2-50000x+240000)(x-50) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}x^5-150x^4+8500x^3-225000x^2+2740000x-12000000\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-10}\right) $ by each term in $ \left( x-20\right) $. $$ \left( \color{blue}{x-10}\right) \cdot \left( x-20\right) = x^2-20x-10x+200 $$ |
| ② | Combine like terms: $$ x^2 \color{blue}{-20x} \color{blue}{-10x} +200 = x^2 \color{blue}{-30x} +200 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-30x+200}\right) $ by each term in $ \left( x-30\right) $. $$ \left( \color{blue}{x^2-30x+200}\right) \cdot \left( x-30\right) = x^3-30x^2-30x^2+900x+200x-6000 $$ |
| ④ | Combine like terms: $$ x^3 \color{blue}{-30x^2} \color{blue}{-30x^2} + \color{red}{900x} + \color{red}{200x} -6000 = x^3 \color{blue}{-60x^2} + \color{red}{1100x} -6000 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{x^3-60x^2+1100x-6000}\right) $ by each term in $ \left( x-40\right) $. $$ \left( \color{blue}{x^3-60x^2+1100x-6000}\right) \cdot \left( x-40\right) = \\ = x^4-40x^3-60x^3+2400x^2+1100x^2-44000x-6000x+240000 $$ |
| ⑥ | Combine like terms: $$ x^4 \color{blue}{-40x^3} \color{blue}{-60x^3} + \color{red}{2400x^2} + \color{red}{1100x^2} \color{green}{-44000x} \color{green}{-6000x} +240000 = \\ = x^4 \color{blue}{-100x^3} + \color{red}{3500x^2} \color{green}{-50000x} +240000 $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{x^4-100x^3+3500x^2-50000x+240000}\right) $ by each term in $ \left( x-50\right) $. $$ \left( \color{blue}{x^4-100x^3+3500x^2-50000x+240000}\right) \cdot \left( x-50\right) = \\ = x^5-50x^4-100x^4+5000x^3+3500x^3-175000x^2-50000x^2+2500000x+240000x-12000000 $$ |
| ⑧ | Combine like terms: $$ x^5 \color{blue}{-50x^4} \color{blue}{-100x^4} + \color{red}{5000x^3} + \color{red}{3500x^3} \color{green}{-175000x^2} \color{green}{-50000x^2} + \color{orange}{2500000x} + \color{orange}{240000x} -12000000 = \\ = x^5 \color{blue}{-150x^4} + \color{red}{8500x^3} \color{green}{-225000x^2} + \color{orange}{2740000x} -12000000 $$ |
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