Question
Answer
$$(100-x)\cdot(99-x)\cdot(98-x)\cdot(97-x)-903450240=x^4-394x^3+58211x^2-3822194x-809340840$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(100-x)\cdot(99-x)\cdot(98-x)\cdot(97-x)-903450240& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(9900-100x-99x+x^2)\cdot(98-x)\cdot(97-x)-903450240 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-199x+9900)\cdot(98-x)\cdot(97-x)-903450240 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(98x^2-x^3-19502x+199x^2+970200-9900x)\cdot(97-x)-903450240 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(-x^3+297x^2-29402x+970200)\cdot(97-x)-903450240 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}x^4-394x^3+58211x^2-3822194x+94109400-903450240 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}x^4-394x^3+58211x^2-3822194x-809340840\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{100-x}\right) $ by each term in $ \left( 99-x\right) $. $$ \left( \color{blue}{100-x}\right) \cdot \left( 99-x\right) = 9900-100x-99x+x^2 $$ |
| ② | Combine like terms: $$ 9900 \color{blue}{-100x} \color{blue}{-99x} +x^2 = x^2 \color{blue}{-199x} +9900 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-199x+9900}\right) $ by each term in $ \left( 98-x\right) $. $$ \left( \color{blue}{x^2-199x+9900}\right) \cdot \left( 98-x\right) = 98x^2-x^3-19502x+199x^2+970200-9900x $$ |
| ④ | Combine like terms: $$ \color{blue}{98x^2} -x^3 \color{red}{-19502x} + \color{blue}{199x^2} +970200 \color{red}{-9900x} = -x^3+ \color{blue}{297x^2} \color{red}{-29402x} +970200 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{-x^3+297x^2-29402x+970200}\right) $ by each term in $ \left( 97-x\right) $. $$ \left( \color{blue}{-x^3+297x^2-29402x+970200}\right) \cdot \left( 97-x\right) = \\ = -97x^3+x^4+28809x^2-297x^3-2851994x+29402x^2+94109400-970200x $$ |
| ⑥ | Combine like terms: $$ \color{blue}{-97x^3} +x^4+ \color{red}{28809x^2} \color{blue}{-297x^3} \color{green}{-2851994x} + \color{red}{29402x^2} +94109400 \color{green}{-970200x} = \\ = x^4 \color{blue}{-394x^3} + \color{red}{58211x^2} \color{green}{-3822194x} +94109400 $$ |
| ⑦ | Combine like terms: $$ x^4-394x^3+58211x^2-3822194x+ \color{blue}{94109400} \color{blue}{-903450240} = x^4-394x^3+58211x^2-3822194x \color{blue}{-809340840} $$ |
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