Question
Answer
$$(m+1)^4-18(m+1)^2(m-1)^2+(m-1)^4=-16m^4+48m^2-16$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(m+1)^4-18(m+1)^2(m-1)^2+(m-1)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}m^4+4m^3+6m^2+4m+1-18(1m^2+2m+1)(1m^2-2m+1)+m^4-4m^3+6m^2-4m+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}m^4+4m^3+6m^2+4m+1-(18m^2+36m+18)(1m^2-2m+1)+m^4-4m^3+6m^2-4m+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}m^4+4m^3+6m^2+4m+1-(18m^4-36m^2+18)+m^4-4m^3+6m^2-4m+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}m^4+4m^3+6m^2+4m+1-18m^4+36m^2-18+m^4-4m^3+6m^2-4m+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}-17m^4+4m^3+42m^2+4m-17+m^4-4m^3+6m^2-4m+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle13}{\textcircled {13}} } }}}-16m^4+48m^2-16\end{aligned} $$ | |
| ① | $$ (m+1)^4 = (m+1)^2 \cdot (m+1)^2 $$ |
| ② | Find $ \left(m+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}{ m } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(m+1\right)^2 = \color{blue}{m^2} +2 \cdot m \cdot 1 + \color{red}{1^2} = m^2+2m+1\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{m^2+2m+1}\right) $ by each term in $ \left( m^2+2m+1\right) $. $$ \left( \color{blue}{m^2+2m+1}\right) \cdot \left( m^2+2m+1\right) = m^4+2m^3+m^2+2m^3+4m^2+2m+m^2+2m+1 $$ |
| ④ | Combine like terms: $$ m^4+ \color{blue}{2m^3} + \color{red}{m^2} + \color{blue}{2m^3} + \color{green}{4m^2} + \color{orange}{2m} + \color{green}{m^2} + \color{orange}{2m} +1 = m^4+ \color{blue}{4m^3} + \color{green}{6m^2} + \color{orange}{4m} +1 $$Find $ \left(m+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}{ m } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(m+1\right)^2 = \color{blue}{m^2} +2 \cdot m \cdot 1 + \color{red}{1^2} = m^2+2m+1\end{aligned} $$Find $ \left(m-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}{ m } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(m-1\right)^2 = \color{blue}{m^2} -2 \cdot m \cdot 1 + \color{red}{1^2} = m^2-2m+1\end{aligned} $$$$ (m-1)^4 = (m-1)^2 \cdot (m-1)^2 $$ |
| ⑤ | Find $ \left(m-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}{ m } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(m-1\right)^2 = \color{blue}{m^2} -2 \cdot m \cdot 1 + \color{red}{1^2} = m^2-2m+1\end{aligned} $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{m^2-2m+1}\right) $ by each term in $ \left( m^2-2m+1\right) $. $$ \left( \color{blue}{m^2-2m+1}\right) \cdot \left( m^2-2m+1\right) = m^4-2m^3+m^2-2m^3+4m^2-2m+m^2-2m+1 $$ |
| ⑦ | Combine like terms: $$ m^4 \color{blue}{-2m^3} + \color{red}{m^2} \color{blue}{-2m^3} + \color{green}{4m^2} \color{orange}{-2m} + \color{green}{m^2} \color{orange}{-2m} +1 = m^4 \color{blue}{-4m^3} + \color{green}{6m^2} \color{orange}{-4m} +1 $$ |
| ⑧ | Multiply $ \color{blue}{18} $ by $ \left( m^2+2m+1\right) $ $$ \color{blue}{18} \cdot \left( m^2+2m+1\right) = 18m^2+36m+18 $$ |
| ⑨ | Multiply each term of $ \left( \color{blue}{18m^2+36m+18}\right) $ by each term in $ \left( m^2-2m+1\right) $. $$ \left( \color{blue}{18m^2+36m+18}\right) \cdot \left( m^2-2m+1\right) = \\ = 18m^4 -\cancel{36m^3}+18m^2+ \cancel{36m^3}-72m^2+ \cancel{36m}+18m^2 -\cancel{36m}+18 $$ |
| ⑩ | Combine like terms: $$ 18m^4 \, \color{blue}{ -\cancel{36m^3}} \,+ \color{green}{18m^2} + \, \color{blue}{ \cancel{36m^3}} \, \color{orange}{-72m^2} + \, \color{blue}{ \cancel{36m}} \,+ \color{orange}{18m^2} \, \color{blue}{ -\cancel{36m}} \,+18 = 18m^4 \color{orange}{-36m^2} +18 $$ |
| ⑪ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 18m^4-36m^2+18 \right) = -18m^4+36m^2-18 $$ |
| ⑫ | Combine like terms: $$ \color{blue}{m^4} +4m^3+ \color{red}{6m^2} +4m+ \color{green}{1} \color{blue}{-18m^4} + \color{red}{36m^2} \color{green}{-18} = \\ = \color{blue}{-17m^4} +4m^3+ \color{red}{42m^2} +4m \color{green}{-17} $$ |
| ⑬ | Combine like terms: $$ \color{blue}{-17m^4} + \, \color{red}{ \cancel{4m^3}} \,+ \color{orange}{42m^2} + \, \color{blue}{ \cancel{4m}} \, \color{green}{-17} + \color{blue}{m^4} \, \color{red}{ -\cancel{4m^3}} \,+ \color{orange}{6m^2} \, \color{blue}{ -\cancel{4m}} \,+ \color{green}{1} = \color{blue}{-16m^4} + \color{orange}{48m^2} \color{green}{-16} $$ |
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