Answer
$$(x-2)^2(x+7)^3=x^5+17x^4+67x^3-161x^2-784x+1372$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-2)^2(x+7)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-4x+4)(x^3+21x^2+147x+343) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^5+17x^4+67x^3-161x^2-784x+1372\end{aligned} $$ | |
| ① | Find $ \left(x-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}{ x } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(x-2\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 2 + \color{red}{2^2} = x^2-4x+4\end{aligned} $$Find $ \left(x+7\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 7 $. $$ \left(x+7\right)^3 = x^3+3 \cdot x^2 \cdot 7 + 3 \cdot x \cdot 7^2+7^3 = x^3+21x^2+147x+343 $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2-4x+4}\right) $ by each term in $ \left( x^3+21x^2+147x+343\right) $. $$ \left( \color{blue}{x^2-4x+4}\right) \cdot \left( x^3+21x^2+147x+343\right) = \\ = x^5+21x^4+147x^3+343x^2-4x^4-84x^3-588x^2-1372x+4x^3+84x^2+588x+1372 $$ |
| ③ | Combine like terms: $$ x^5+ \color{blue}{21x^4} + \color{red}{147x^3} + \color{green}{343x^2} \color{blue}{-4x^4} \color{orange}{-84x^3} \color{blue}{-588x^2} \color{red}{-1372x} + \color{orange}{4x^3} + \color{blue}{84x^2} + \color{red}{588x} +1372 = \\ = x^5+ \color{blue}{17x^4} + \color{orange}{67x^3} \color{blue}{-161x^2} \color{red}{-784x} +1372 $$ |
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