Question
Answer
$$x(x-1)(x(x-1)(x-2)(x-3)(x-3)+2x(x-1)(x-2)+1)=x^7-10x^6+40x^5-76x^4+67x^3-21x^2-x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x(x-1)(x(x-1)(x-2)(x-3)(x-3)+2x(x-1)(x-2)+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-x)((x^2-x)(x-2)(x-3)(x-3)+(2x^2-2x)(x-2)+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-x)((x^3-2x^2-x^2+2x)(x-3)(x-3)+2x^3-4x^2-2x^2+4x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^2-x)((x^3-3x^2+2x)(x-3)(x-3)+2x^3-6x^2+4x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^2-x)((x^4-3x^3-3x^3+9x^2+2x^2-6x)(x-3)+2x^3-6x^2+4x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(x^2-x)((x^4-6x^3+11x^2-6x)(x-3)+2x^3-6x^2+4x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}(x^2-x)(x^5-9x^4+29x^3-39x^2+18x+2x^3-6x^2+4x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}(x^2-x)(x^5-9x^4+31x^3-45x^2+22x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}x^7-10x^6+40x^5-76x^4+67x^3-21x^2-x\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{x} $ by $ \left( x-1\right) $ $$ \color{blue}{x} \cdot \left( x-1\right) = x^2-x $$Multiply $ \color{blue}{x} $ by $ \left( x-1\right) $ $$ \color{blue}{x} \cdot \left( x-1\right) = x^2-x $$Multiply $ \color{blue}{2x} $ by $ \left( x-1\right) $ $$ \color{blue}{2x} \cdot \left( x-1\right) = 2x^2-2x $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2-x}\right) $ by each term in $ \left( x-2\right) $. $$ \left( \color{blue}{x^2-x}\right) \cdot \left( x-2\right) = x^3-2x^2-x^2+2x $$Multiply each term of $ \left( \color{blue}{2x^2-2x}\right) $ by each term in $ \left( x-2\right) $. $$ \left( \color{blue}{2x^2-2x}\right) \cdot \left( x-2\right) = 2x^3-4x^2-2x^2+4x $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-2x^2} \color{blue}{-x^2} +2x = x^3 \color{blue}{-3x^2} +2x $$Combine like terms: $$ 2x^3 \color{blue}{-4x^2} \color{blue}{-2x^2} +4x = 2x^3 \color{blue}{-6x^2} +4x $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-3x^2+2x}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{x^3-3x^2+2x}\right) \cdot \left( x-3\right) = x^4-3x^3-3x^3+9x^2+2x^2-6x $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-3x^3} \color{blue}{-3x^3} + \color{red}{9x^2} + \color{red}{2x^2} -6x = x^4 \color{blue}{-6x^3} + \color{red}{11x^2} -6x $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{x^4-6x^3+11x^2-6x}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{x^4-6x^3+11x^2-6x}\right) \cdot \left( x-3\right) = x^5-3x^4-6x^4+18x^3+11x^3-33x^2-6x^2+18x $$ |
| ⑦ | Combine like terms: $$ x^5 \color{blue}{-3x^4} \color{blue}{-6x^4} + \color{red}{18x^3} + \color{red}{11x^3} \color{green}{-33x^2} \color{green}{-6x^2} +18x = \\ = x^5 \color{blue}{-9x^4} + \color{red}{29x^3} \color{green}{-39x^2} +18x $$ |
| ⑧ | Combine like terms: $$ x^5-9x^4+ \color{blue}{29x^3} \color{red}{-39x^2} + \color{green}{18x} + \color{blue}{2x^3} \color{red}{-6x^2} + \color{green}{4x} = \\ = x^5-9x^4+ \color{blue}{31x^3} \color{red}{-45x^2} + \color{green}{22x} $$ |
| ⑨ | Multiply each term of $ \left( \color{blue}{x^2-x}\right) $ by each term in $ \left( x^5-9x^4+31x^3-45x^2+22x+1\right) $. $$ \left( \color{blue}{x^2-x}\right) \cdot \left( x^5-9x^4+31x^3-45x^2+22x+1\right) = \\ = x^7-9x^6+31x^5-45x^4+22x^3+x^2-x^6+9x^5-31x^4+45x^3-22x^2-x $$ |
| ⑩ | Combine like terms: $$ x^7 \color{blue}{-9x^6} + \color{red}{31x^5} \color{green}{-45x^4} + \color{orange}{22x^3} + \color{blue}{x^2} \color{blue}{-x^6} + \color{red}{9x^5} \color{green}{-31x^4} + \color{orange}{45x^3} \color{blue}{-22x^2} -x = \\ = x^7 \color{blue}{-10x^6} + \color{red}{40x^5} \color{green}{-76x^4} + \color{orange}{67x^3} \color{blue}{-21x^2} -x $$ |
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