Question
Answer
$$(x+2)(x-3)^2(x-1)=x^4-5x^3+x^2+21x-18$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+2)(x-3)^2(x-1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x+2)(x^2-6x+9)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-6x^2+9x+2x^2-12x+18)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-4x^2-3x+18)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-5x^3+x^2+21x-18\end{aligned} $$ | |
| ① | 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}\right) $ by each term in $ \left( x^2-6x+9\right) $. $$ \left( \color{blue}{x+2}\right) \cdot \left( x^2-6x+9\right) = x^3-6x^2+9x+2x^2-12x+18 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-6x^2} + \color{red}{9x} + \color{blue}{2x^2} \color{red}{-12x} +18 = x^3 \color{blue}{-4x^2} \color{red}{-3x} +18 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-4x^2-3x+18}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{x^3-4x^2-3x+18}\right) \cdot \left( x-1\right) = x^4-x^3-4x^3+4x^2-3x^2+3x+18x-18 $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-x^3} \color{blue}{-4x^3} + \color{red}{4x^2} \color{red}{-3x^2} + \color{green}{3x} + \color{green}{18x} -18 = x^4 \color{blue}{-5x^3} + \color{red}{x^2} + \color{green}{21x} -18 $$ |
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