Question
Answer
$$(x+7)^2(x+1)(x-3)=x^4+12x^3+18x^2-140x-147$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+7)^2(x+1)(x-3)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+14x+49)(x+1)(x-3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3+x^2+14x^2+14x+49x+49)(x-3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3+15x^2+63x+49)(x-3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4+12x^3+18x^2-140x-147\end{aligned} $$ | |
| ① | Find $ \left(x+7\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}{ 7 }$. $$ \begin{aligned}\left(x+7\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 7 + \color{red}{7^2} = x^2+14x+49\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2+14x+49}\right) $ by each term in $ \left( x+1\right) $. $$ \left( \color{blue}{x^2+14x+49}\right) \cdot \left( x+1\right) = x^3+x^2+14x^2+14x+49x+49 $$ |
| ③ | Combine like terms: $$ x^3+ \color{blue}{x^2} + \color{blue}{14x^2} + \color{red}{14x} + \color{red}{49x} +49 = x^3+ \color{blue}{15x^2} + \color{red}{63x} +49 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3+15x^2+63x+49}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{x^3+15x^2+63x+49}\right) \cdot \left( x-3\right) = x^4-3x^3+15x^3-45x^2+63x^2-189x+49x-147 $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-3x^3} + \color{blue}{15x^3} \color{red}{-45x^2} + \color{red}{63x^2} \color{green}{-189x} + \color{green}{49x} -147 = \\ = x^4+ \color{blue}{12x^3} + \color{red}{18x^2} \color{green}{-140x} -147 $$ |
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