Question
Answer
$$(x+2)^2(x-3)^2(x-1)=x^5-3x^4-9x^3+23x^2+24x-36$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+2)^2(x-3)^2(x-1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+4x+4)(x^2-6x+9)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^4-2x^3-11x^2+12x+36)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^5-3x^4-9x^3+23x^2+24x-36\end{aligned} $$ | |
| 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-3\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}{ 3 }$. $$ \begin{aligned}\left(x-3\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 3 + \color{red}{3^2} = x^2-6x+9\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2+4x+4}\right) $ by each term in $ \left( x^2-6x+9\right) $. $$ \left( \color{blue}{x^2+4x+4}\right) \cdot \left( x^2-6x+9\right) = x^4-6x^3+9x^2+4x^3-24x^2+36x+4x^2-24x+36 $$ |
| ③ | Combine like terms: $$ x^4 \color{blue}{-6x^3} + \color{red}{9x^2} + \color{blue}{4x^3} \color{green}{-24x^2} + \color{orange}{36x} + \color{green}{4x^2} \color{orange}{-24x} +36 = \\ = x^4 \color{blue}{-2x^3} \color{green}{-11x^2} + \color{orange}{12x} +36 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^4-2x^3-11x^2+12x+36}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{x^4-2x^3-11x^2+12x+36}\right) \cdot \left( x-1\right) = x^5-x^4-2x^4+2x^3-11x^3+11x^2+12x^2-12x+36x-36 $$ |
| ⑤ | Combine like terms: $$ x^5 \color{blue}{-x^4} \color{blue}{-2x^4} + \color{red}{2x^3} \color{red}{-11x^3} + \color{green}{11x^2} + \color{green}{12x^2} \color{orange}{-12x} + \color{orange}{36x} -36 = \\ = x^5 \color{blue}{-3x^4} \color{red}{-9x^3} + \color{green}{23x^2} + \color{orange}{24x} -36 $$ |
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