Question
Answer
$$108t^7+(9t^2+1)^2-(3t^2+1)^2(12t^2+1)=108t^7-108t^6$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}108t^7+(9t^2+1)^2-(3t^2+1)^2(12t^2+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}108t^7+81t^4+18t^2+1-(9t^4+6t^2+1)(12t^2+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}108t^7+81t^4+18t^2+1-(108t^6+9t^4+72t^4+6t^2+12t^2+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}108t^7+81t^4+18t^2+1-(108t^6+81t^4+18t^2+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}108t^7+81t^4+18t^2+1-108t^6-81t^4-18t^2-1 \xlongequal{ } \\[1 em] & \xlongequal{ }108t^7+ \cancel{81t^4}+ \cancel{18t^2}+ \cancel{1}-108t^6 -\cancel{81t^4} -\cancel{18t^2} -\cancel{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}108t^7-108t^6\end{aligned} $$ | |
| ① | Find $ \left(9t^2+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}{ 9t^2 } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(9t^2+1\right)^2 = \color{blue}{\left( 9t^2 \right)^2} +2 \cdot 9t^2 \cdot 1 + \color{red}{1^2} = 81t^4+18t^2+1\end{aligned} $$Find $ \left(3t^2+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}{ 3t^2 } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(3t^2+1\right)^2 = \color{blue}{\left( 3t^2 \right)^2} +2 \cdot 3t^2 \cdot 1 + \color{red}{1^2} = 9t^4+6t^2+1\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{9t^4+6t^2+1}\right) $ by each term in $ \left( 12t^2+1\right) $. $$ \left( \color{blue}{9t^4+6t^2+1}\right) \cdot \left( 12t^2+1\right) = 108t^6+9t^4+72t^4+6t^2+12t^2+1 $$ |
| ③ | Combine like terms: $$ 108t^7+81t^4+18t^2+1 = 108t^7+81t^4+18t^2+1 $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 108t^6+81t^4+18t^2+1 \right) = -108t^6-81t^4-18t^2-1 $$ |
| ⑤ | Combine like terms: $$ 108t^7+ \, \color{blue}{ \cancel{81t^4}} \,+ \, \color{green}{ \cancel{18t^2}} \,+ \, \color{blue}{ \cancel{1}} \,-108t^6 \, \color{blue}{ -\cancel{81t^4}} \, \, \color{green}{ -\cancel{18t^2}} \, \, \color{blue}{ -\cancel{1}} \, = 108t^7-108t^6 $$ |
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