Answer
$$(y+3)(y+3)(y+3)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y)=-y^8+y^7+23y^6-37y^5-170y^4+392y^3+288y^2-1296y+864$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(y+3)(y+3)(y+3)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1y^2+3y+3y+9)(y+3)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1y^2+6y+9)(y+3)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(1y^3+3y^2+6y^2+18y+9y+27)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(1y^3+9y^2+27y+27)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(-y^4-7y^3-9y^2+27y+54)\cdot(2-y)\cdot(2-y)\cdot(2-y)\cdot(2-y) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}(1y^5+5y^4-5y^3-45y^2+108)\cdot(2-y)\cdot(2-y)\cdot(2-y) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}(-y^6-3y^5+15y^4+35y^3-90y^2-108y+216)\cdot(2-y)\cdot(2-y) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}(1y^7+y^6-21y^5-5y^4+160y^3-72y^2-432y+432)\cdot(2-y) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle13}{\textcircled {13}} \htmlClass{explanationCircle explanationCircle14}{\textcircled {14}} } }}}-y^8+y^7+23y^6-37y^5-170y^4+392y^3+288y^2-1296y+864\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{y+3}\right) $ by each term in $ \left( y+3\right) $. $$ \left( \color{blue}{y+3}\right) \cdot \left( y+3\right) = y^2+3y+3y+9 $$ |
| ② | Combine like terms: $$ y^2+ \color{blue}{3y} + \color{blue}{3y} +9 = y^2+ \color{blue}{6y} +9 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{y^2+6y+9}\right) $ by each term in $ \left( y+3\right) $. $$ \left( \color{blue}{y^2+6y+9}\right) \cdot \left( y+3\right) = y^3+3y^2+6y^2+18y+9y+27 $$ |
| ④ | Combine like terms: $$ y^3+ \color{blue}{3y^2} + \color{blue}{6y^2} + \color{red}{18y} + \color{red}{9y} +27 = y^3+ \color{blue}{9y^2} + \color{red}{27y} +27 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{y^3+9y^2+27y+27}\right) $ by each term in $ \left( 2-y\right) $. $$ \left( \color{blue}{y^3+9y^2+27y+27}\right) \cdot \left( 2-y\right) = 2y^3-y^4+18y^2-9y^3+54y-27y^2+54-27y $$ |
| ⑥ | Combine like terms: $$ \color{blue}{2y^3} -y^4+ \color{red}{18y^2} \color{blue}{-9y^3} + \color{green}{54y} \color{red}{-27y^2} +54 \color{green}{-27y} = -y^4 \color{blue}{-7y^3} \color{red}{-9y^2} + \color{green}{27y} +54 $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{-y^4-7y^3-9y^2+27y+54}\right) $ by each term in $ \left( 2-y\right) $. $$ \left( \color{blue}{-y^4-7y^3-9y^2+27y+54}\right) \cdot \left( 2-y\right) = \\ = -2y^4+y^5-14y^3+7y^4-18y^2+9y^3+ \cancel{54y}-27y^2+108 -\cancel{54y} $$ |
| ⑧ | Combine like terms: $$ \color{blue}{-2y^4} +y^5 \color{red}{-14y^3} + \color{blue}{7y^4} \color{green}{-18y^2} + \color{red}{9y^3} + \, \color{orange}{ \cancel{54y}} \, \color{green}{-27y^2} +108 \, \color{orange}{ -\cancel{54y}} \, = \\ = y^5+ \color{blue}{5y^4} \color{red}{-5y^3} \color{green}{-45y^2} +108 $$ |
| ⑨ | Multiply each term of $ \left( \color{blue}{y^5+5y^4-5y^3-45y^2+108}\right) $ by each term in $ \left( 2-y\right) $. $$ \left( \color{blue}{y^5+5y^4-5y^3-45y^2+108}\right) \cdot \left( 2-y\right) = \\ = 2y^5-y^6+10y^4-5y^5-10y^3+5y^4-90y^2+45y^3+216-108y $$ |
| ⑩ | Combine like terms: $$ \color{blue}{2y^5} -y^6+ \color{red}{10y^4} \color{blue}{-5y^5} \color{green}{-10y^3} + \color{red}{5y^4} -90y^2+ \color{green}{45y^3} +216-108y = \\ = -y^6 \color{blue}{-3y^5} + \color{red}{15y^4} + \color{green}{35y^3} -90y^2-108y+216 $$ |
| ⑪ | Multiply each term of $ \left( \color{blue}{-y^6-3y^5+15y^4+35y^3-90y^2-108y+216}\right) $ by each term in $ \left( 2-y\right) $. $$ \left( \color{blue}{-y^6-3y^5+15y^4+35y^3-90y^2-108y+216}\right) \cdot \left( 2-y\right) = \\ = -2y^6+y^7-6y^5+3y^6+30y^4-15y^5+70y^3-35y^4-180y^2+90y^3-216y+108y^2+432-216y $$ |
| ⑫ | Combine like terms: $$ \color{blue}{-2y^6} +y^7 \color{red}{-6y^5} + \color{blue}{3y^6} + \color{green}{30y^4} \color{red}{-15y^5} + \color{orange}{70y^3} \color{green}{-35y^4} \color{blue}{-180y^2} + \color{orange}{90y^3} \color{red}{-216y} + \color{blue}{108y^2} +432 \color{red}{-216y} = \\ = y^7+ \color{blue}{y^6} \color{red}{-21y^5} \color{green}{-5y^4} + \color{orange}{160y^3} \color{blue}{-72y^2} \color{red}{-432y} +432 $$ |
| ⑬ | Multiply each term of $ \left( \color{blue}{y^7+y^6-21y^5-5y^4+160y^3-72y^2-432y+432}\right) $ by each term in $ \left( 2-y\right) $. $$ \left( \color{blue}{y^7+y^6-21y^5-5y^4+160y^3-72y^2-432y+432}\right) \cdot \left( 2-y\right) = \\ = 2y^7-y^8+2y^6-y^7-42y^5+21y^6-10y^4+5y^5+320y^3-160y^4-144y^2+72y^3-864y+432y^2+864-432y $$ |
| ⑭ | Combine like terms: $$ \color{blue}{2y^7} -y^8+ \color{red}{2y^6} \color{blue}{-y^7} \color{green}{-42y^5} + \color{red}{21y^6} \color{orange}{-10y^4} + \color{green}{5y^5} + \color{blue}{320y^3} \color{orange}{-160y^4} \color{red}{-144y^2} + \color{blue}{72y^3} \color{green}{-864y} + \color{red}{432y^2} +864 \color{green}{-432y} = \\ = -y^8+ \color{blue}{y^7} + \color{red}{23y^6} \color{green}{-37y^5} \color{orange}{-170y^4} + \color{blue}{392y^3} + \color{red}{288y^2} \color{green}{-1296y} +864 $$ |
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