Question
Answer
$$(x-2)(x-2)(x-2)(x-2)(x+1)(x+1)=x^6-6x^5+9x^4+8x^3-24x^2+16$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-2)(x-2)(x-2)(x-2)(x+1)(x+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-2x-2x+4)(x-2)(x-2)(x+1)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-4x+4)(x-2)(x-2)(x+1)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-2x^2-4x^2+8x+4x-8)(x-2)(x+1)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^3-6x^2+12x-8)(x-2)(x+1)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(x^4-8x^3+24x^2-32x+16)(x+1)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}(x^5-7x^4+16x^3-8x^2-16x+16)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}x^6-6x^5+9x^4+8x^3-24x^2+16\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-2}\right) $ by each term in $ \left( x-2\right) $. $$ \left( \color{blue}{x-2}\right) \cdot \left( x-2\right) = x^2-2x-2x+4 $$ |
| ② | Combine like terms: $$ x^2 \color{blue}{-2x} \color{blue}{-2x} +4 = x^2 \color{blue}{-4x} +4 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-4x+4}\right) $ by each term in $ \left( x-2\right) $. $$ \left( \color{blue}{x^2-4x+4}\right) \cdot \left( x-2\right) = x^3-2x^2-4x^2+8x+4x-8 $$ |
| ④ | Combine like terms: $$ x^3 \color{blue}{-2x^2} \color{blue}{-4x^2} + \color{red}{8x} + \color{red}{4x} -8 = x^3 \color{blue}{-6x^2} + \color{red}{12x} -8 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{x^3-6x^2+12x-8}\right) $ by each term in $ \left( x-2\right) $. $$ \left( \color{blue}{x^3-6x^2+12x-8}\right) \cdot \left( x-2\right) = x^4-2x^3-6x^3+12x^2+12x^2-24x-8x+16 $$ |
| ⑥ | Combine like terms: $$ x^4 \color{blue}{-2x^3} \color{blue}{-6x^3} + \color{red}{12x^2} + \color{red}{12x^2} \color{green}{-24x} \color{green}{-8x} +16 = x^4 \color{blue}{-8x^3} + \color{red}{24x^2} \color{green}{-32x} +16 $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{x^4-8x^3+24x^2-32x+16}\right) $ by each term in $ \left( x+1\right) $. $$ \left( \color{blue}{x^4-8x^3+24x^2-32x+16}\right) \cdot \left( x+1\right) = x^5+x^4-8x^4-8x^3+24x^3+24x^2-32x^2-32x+16x+16 $$ |
| ⑧ | Combine like terms: $$ x^5+ \color{blue}{x^4} \color{blue}{-8x^4} \color{red}{-8x^3} + \color{red}{24x^3} + \color{green}{24x^2} \color{green}{-32x^2} \color{orange}{-32x} + \color{orange}{16x} +16 = \\ = x^5 \color{blue}{-7x^4} + \color{red}{16x^3} \color{green}{-8x^2} \color{orange}{-16x} +16 $$ |
| ⑨ | Multiply each term of $ \left( \color{blue}{x^5-7x^4+16x^3-8x^2-16x+16}\right) $ by each term in $ \left( x+1\right) $. $$ \left( \color{blue}{x^5-7x^4+16x^3-8x^2-16x+16}\right) \cdot \left( x+1\right) = \\ = x^6+x^5-7x^5-7x^4+16x^4+16x^3-8x^3-8x^2-16x^2 -\cancel{16x}+ \cancel{16x}+16 $$ |
| ⑩ | Combine like terms: $$ x^6+ \color{blue}{x^5} \color{blue}{-7x^5} \color{red}{-7x^4} + \color{red}{16x^4} + \color{green}{16x^3} \color{green}{-8x^3} \color{orange}{-8x^2} \color{orange}{-16x^2} \, \color{blue}{ -\cancel{16x}} \,+ \, \color{blue}{ \cancel{16x}} \,+16 = \\ = x^6 \color{blue}{-6x^5} + \color{red}{9x^4} + \color{green}{8x^3} \color{orange}{-24x^2} +16 $$ |
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