Question
Answer
$$(t+2)^3-6(t+2)^2+11(t+2)-6=t^3-t$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(t+2)^3-6(t+2)^2+11(t+2)-6& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}t^3+6t^2+12t+8-6(1t^2+4t+4)+11(t+2)-6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}t^3+6t^2+12t+8-(6t^2+24t+24)+11t+22-6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}t^3+6t^2+12t+8-6t^2-24t-24+11t+22-6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}t^3-12t-16+11t+22-6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}t^3-t+6-6 \xlongequal{ } \\[1 em] & \xlongequal{ }t^3-t+ \cancel{6} -\cancel{6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}t^3-t\end{aligned} $$ | |
| ① | Find $ \left(t+2\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = t $ and $ B = 2 $. $$ \left(t+2\right)^3 = t^3+3 \cdot t^2 \cdot 2 + 3 \cdot t \cdot 2^2+2^3 = t^3+6t^2+12t+8 $$Find $ \left(t+2\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ t } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(t+2\right)^2 = \color{blue}{t^2} +2 \cdot t \cdot 2 + \color{red}{2^2} = t^2+4t+4\end{aligned} $$ |
| ② | Multiply $ \color{blue}{6} $ by $ \left( t^2+4t+4\right) $ $$ \color{blue}{6} \cdot \left( t^2+4t+4\right) = 6t^2+24t+24 $$Multiply $ \color{blue}{11} $ by $ \left( t+2\right) $ $$ \color{blue}{11} \cdot \left( t+2\right) = 11t+22 $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 6t^2+24t+24 \right) = -6t^2-24t-24 $$ |
| ④ | Combine like terms: $$ t^3+ \, \color{blue}{ \cancel{6t^2}} \,+ \color{green}{12t} + \color{orange}{8} \, \color{blue}{ -\cancel{6t^2}} \, \color{green}{-24t} \color{orange}{-24} = t^3 \color{green}{-12t} \color{orange}{-16} $$ |
| ⑤ | Combine like terms: $$ t^3 \color{blue}{-12t} \color{red}{-16} + \color{blue}{11t} + \color{red}{22} = t^3 \color{blue}{-t} + \color{red}{6} $$ |
| ⑥ | Combine like terms: $$ t^3-t+ \, \color{blue}{ \cancel{6}} \, \, \color{blue}{ -\cancel{6}} \, = t^3-t $$ |
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