Question
Answer
$$(5-t)^2(4-t)^2=t^4-18t^3+121t^2-360t+400$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(5-t)^2(4-t)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(25-10t+t^2)(16-8t+t^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}t^4-18t^3+121t^2-360t+400\end{aligned} $$ | |
| ① | Find $ \left(5-t\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 5 } $ and $ B = \color{red}{ t }$. $$ \begin{aligned}\left(5-t\right)^2 = \color{blue}{5^2} -2 \cdot 5 \cdot t + \color{red}{t^2} = 25-10t+t^2\end{aligned} $$Find $ \left(4-t\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 4 } $ and $ B = \color{red}{ t }$. $$ \begin{aligned}\left(4-t\right)^2 = \color{blue}{4^2} -2 \cdot 4 \cdot t + \color{red}{t^2} = 16-8t+t^2\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{25-10t+t^2}\right) $ by each term in $ \left( 16-8t+t^2\right) $. $$ \left( \color{blue}{25-10t+t^2}\right) \cdot \left( 16-8t+t^2\right) = 400-200t+25t^2-160t+80t^2-10t^3+16t^2-8t^3+t^4 $$ |
| ③ | Combine like terms: $$ 400 \color{blue}{-200t} + \color{red}{25t^2} \color{blue}{-160t} + \color{green}{80t^2} \color{orange}{-10t^3} + \color{green}{16t^2} \color{orange}{-8t^3} +t^4 = \\ = t^4 \color{orange}{-18t^3} + \color{green}{121t^2} \color{blue}{-360t} +400 $$ |
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