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