Question
Answer
$$x(x-1)^2(x-2)^2((x-2)(x-1)+1)=x^7-9x^6+34x^5-69x^4+79x^3-48x^2+12x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x(x-1)^2(x-2)^2((x-2)(x-1)+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x(x^2-2x+1)(x^2-4x+4)((x-2)(x-1)+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-2x^2+x)(x^2-4x+4)(x^2-x-2x+2+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-2x^2+x)(x^2-4x+4)(x^2-3x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(x^5-6x^4+13x^3-12x^2+4x)(x^2-3x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}x^7-9x^6+34x^5-69x^4+79x^3-48x^2+12x\end{aligned} $$ | |
| ① | Find $ \left(x-1\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x-1\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 1 + \color{red}{1^2} = x^2-2x+1\end{aligned} $$Find $ \left(x-2\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(x-2\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 2 + \color{red}{2^2} = x^2-4x+4\end{aligned} $$ |
| ② | Multiply $ \color{blue}{x} $ by $ \left( x^2-2x+1\right) $ $$ \color{blue}{x} \cdot \left( x^2-2x+1\right) = x^3-2x^2+x $$ Multiply each term of $ \left( \color{blue}{x-2}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{x-2}\right) \cdot \left( x-1\right) = x^2-x-2x+2 $$ |
| ③ | Combine like terms: $$ x^2 \color{blue}{-x} \color{blue}{-2x} + \color{red}{2} + \color{red}{1} = x^2 \color{blue}{-3x} + \color{red}{3} $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-2x^2+x}\right) $ by each term in $ \left( x^2-4x+4\right) $. $$ \left( \color{blue}{x^3-2x^2+x}\right) \cdot \left( x^2-4x+4\right) = x^5-4x^4+4x^3-2x^4+8x^3-8x^2+x^3-4x^2+4x $$ |
| ⑤ | Combine like terms: $$ x^5 \color{blue}{-4x^4} + \color{red}{4x^3} \color{blue}{-2x^4} + \color{green}{8x^3} \color{orange}{-8x^2} + \color{green}{x^3} \color{orange}{-4x^2} +4x = \\ = x^5 \color{blue}{-6x^4} + \color{green}{13x^3} \color{orange}{-12x^2} +4x $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{x^5-6x^4+13x^3-12x^2+4x}\right) $ by each term in $ \left( x^2-3x+3\right) $. $$ \left( \color{blue}{x^5-6x^4+13x^3-12x^2+4x}\right) \cdot \left( x^2-3x+3\right) = \\ = x^7-3x^6+3x^5-6x^6+18x^5-18x^4+13x^5-39x^4+39x^3-12x^4+36x^3-36x^2+4x^3-12x^2+12x $$ |
| ⑦ | Combine like terms: $$ x^7 \color{blue}{-3x^6} + \color{red}{3x^5} \color{blue}{-6x^6} + \color{green}{18x^5} \color{orange}{-18x^4} + \color{green}{13x^5} \color{blue}{-39x^4} + \color{red}{39x^3} \color{blue}{-12x^4} + \color{green}{36x^3} \color{orange}{-36x^2} + \color{green}{4x^3} \color{orange}{-12x^2} +12x = \\ = x^7 \color{blue}{-9x^6} + \color{green}{34x^5} \color{blue}{-69x^4} + \color{green}{79x^3} \color{orange}{-48x^2} +12x $$ |
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