Question
Answer
$$x(-x+1)(x+4)(x-7)=-x^4+4x^3+25x^2-28x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x(-x+1)(x+4)(x-7)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(-x^2+x)(x+4)(x-7) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(-x^3-4x^2+x^2+4x)(x-7) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(-x^3-3x^2+4x)(x-7) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-x^4+7x^3-3x^3+21x^2+4x^2-28x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-x^4+4x^3+25x^2-28x\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{x} $ by $ \left( -x+1\right) $ $$ \color{blue}{x} \cdot \left( -x+1\right) = -x^2+x $$ |
| ② | Multiply each term of $ \left( \color{blue}{-x^2+x}\right) $ by each term in $ \left( x+4\right) $. $$ \left( \color{blue}{-x^2+x}\right) \cdot \left( x+4\right) = -x^3-4x^2+x^2+4x $$ |
| ③ | Combine like terms: $$ -x^3 \color{blue}{-4x^2} + \color{blue}{x^2} +4x = -x^3 \color{blue}{-3x^2} +4x $$ |
| ④ | Multiply each term of $ \left( \color{blue}{-x^3-3x^2+4x}\right) $ by each term in $ \left( x-7\right) $. $$ \left( \color{blue}{-x^3-3x^2+4x}\right) \cdot \left( x-7\right) = -x^4+7x^3-3x^3+21x^2+4x^2-28x $$ |
| ⑤ | Combine like terms: $$ -x^4+ \color{blue}{7x^3} \color{blue}{-3x^3} + \color{red}{21x^2} + \color{red}{4x^2} -28x = -x^4+ \color{blue}{4x^3} + \color{red}{25x^2} -28x $$ |
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