Question
Answer
$$(r^2-2r+2)^3r(r-1)^2=r^9-8r^8+31r^7-74r^6+118r^5-128r^4+92r^3-40r^2+8r$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(r^2-2r+2)^3r(r-1)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(1r^6-6r^5+18r^4-32r^3+36r^2-24r+8)r(1r^2-2r+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(1r^7-6r^6+18r^5-32r^4+36r^3-24r^2+8r)(1r^2-2r+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}r^9-8r^8+31r^7-74r^6+118r^5-128r^4+92r^3-40r^2+8r\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{r^2-2r+2}\right) $ by each term in $ \left( r^2-2r+2\right) $. $$ \left( \color{blue}{r^2-2r+2}\right) \cdot \left( r^2-2r+2\right) = r^4-2r^3+2r^2-2r^3+4r^2-4r+2r^2-4r+4 $$ |
| ② | Combine like terms: $$ r^4 \color{blue}{-2r^3} + \color{red}{2r^2} \color{blue}{-2r^3} + \color{green}{4r^2} \color{orange}{-4r} + \color{green}{2r^2} \color{orange}{-4r} +4 = r^4 \color{blue}{-4r^3} + \color{green}{8r^2} \color{orange}{-8r} +4 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{r^4-4r^3+8r^2-8r+4}\right) $ by each term in $ \left( r^2-2r+2\right) $. $$ \left( \color{blue}{r^4-4r^3+8r^2-8r+4}\right) \cdot \left( r^2-2r+2\right) = \\ = r^6-2r^5+2r^4-4r^5+8r^4-8r^3+8r^4-16r^3+16r^2-8r^3+16r^2-16r+4r^2-8r+8 $$ |
| ④ | Combine like terms: $$ r^6 \color{blue}{-2r^5} + \color{red}{2r^4} \color{blue}{-4r^5} + \color{green}{8r^4} \color{orange}{-8r^3} + \color{green}{8r^4} \color{blue}{-16r^3} + \color{red}{16r^2} \color{blue}{-8r^3} + \color{green}{16r^2} \color{orange}{-16r} + \color{green}{4r^2} \color{orange}{-8r} +8 = \\ = r^6 \color{blue}{-6r^5} + \color{green}{18r^4} \color{blue}{-32r^3} + \color{green}{36r^2} \color{orange}{-24r} +8 $$Find $ \left(r-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}{ r } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(r-1\right)^2 = \color{blue}{r^2} -2 \cdot r \cdot 1 + \color{red}{1^2} = r^2-2r+1\end{aligned} $$ |
| ⑤ | $$ \left( \color{blue}{r^6-6r^5+18r^4-32r^3+36r^2-24r+8}\right) \cdot r = r^7-6r^6+18r^5-32r^4+36r^3-24r^2+8r $$ |
| ⑥ | Combine like terms: $$ r^7-6r^6+18r^5-32r^4+36r^3-24r^2+8r = r^7-6r^6+18r^5-32r^4+36r^3-24r^2+8r $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{r^7-6r^6+18r^5-32r^4+36r^3-24r^2+8r}\right) $ by each term in $ \left( r^2-2r+1\right) $. $$ \left( \color{blue}{r^7-6r^6+18r^5-32r^4+36r^3-24r^2+8r}\right) \cdot \left( r^2-2r+1\right) = \\ = r^9-2r^8+r^7-6r^8+12r^7-6r^6+18r^7-36r^6+18r^5-32r^6+64r^5-32r^4+36r^5-72r^4+36r^3-24r^4+48r^3-24r^2+8r^3-16r^2+8r $$ |
| ⑧ | Combine like terms: $$ r^9 \color{blue}{-2r^8} + \color{red}{r^7} \color{blue}{-6r^8} + \color{green}{12r^7} \color{orange}{-6r^6} + \color{green}{18r^7} \color{blue}{-36r^6} + \color{red}{18r^5} \color{blue}{-32r^6} + \color{green}{64r^5} \color{orange}{-32r^4} + \color{green}{36r^5} \color{blue}{-72r^4} + \color{red}{36r^3} \color{blue}{-24r^4} + \color{green}{48r^3} \color{orange}{-24r^2} + \color{green}{8r^3} \color{orange}{-16r^2} +8r = \\ = r^9 \color{blue}{-8r^8} + \color{green}{31r^7} \color{blue}{-74r^6} + \color{green}{118r^5} \color{blue}{-128r^4} + \color{green}{92r^3} \color{orange}{-40r^2} +8r $$ |
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