Question
Answer
$$(x-3)(x+1)^2(x+2)=x^4+x^3-7x^2-13x-6$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-3)(x+1)^2(x+2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x-3)(x^2+2x+1)(x+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3+2x^2+x-3x^2-6x-3)(x+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-x^2-5x-3)(x+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4+x^3-7x^2-13x-6\end{aligned} $$ | |
| ① | Find $ \left(x+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}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x-3}\right) $ by each term in $ \left( x^2+2x+1\right) $. $$ \left( \color{blue}{x-3}\right) \cdot \left( x^2+2x+1\right) = x^3+2x^2+x-3x^2-6x-3 $$ |
| ③ | Combine like terms: $$ x^3+ \color{blue}{2x^2} + \color{red}{x} \color{blue}{-3x^2} \color{red}{-6x} -3 = x^3 \color{blue}{-x^2} \color{red}{-5x} -3 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-x^2-5x-3}\right) $ by each term in $ \left( x+2\right) $. $$ \left( \color{blue}{x^3-x^2-5x-3}\right) \cdot \left( x+2\right) = x^4+2x^3-x^3-2x^2-5x^2-10x-3x-6 $$ |
| ⑤ | Combine like terms: $$ x^4+ \color{blue}{2x^3} \color{blue}{-x^3} \color{red}{-2x^2} \color{red}{-5x^2} \color{green}{-10x} \color{green}{-3x} -6 = x^4+ \color{blue}{x^3} \color{red}{-7x^2} \color{green}{-13x} -6 $$ |
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