Question
Answer
$$(5-x)\cdot(1-x)\cdot(-7-x)+2\cdot(7+x)-20+40\cdot(1-x)=-x^3-x^2-x-1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(5-x)\cdot(1-x)\cdot(-7-x)+2\cdot(7+x)-20+40\cdot(1-x)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(5-5x-x+x^2)\cdot(-7-x)+14+2x-20+40-40x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-6x+5)\cdot(-7-x)+14+2x-20+40-40x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-7x^2-x^3+42x+6x^2-35-5x+14+2x-20+40-40x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-x^3-x^2+37x-35+14+2x-20+40-40x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-x^3-x^2+39x-21-20+40-40x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-x^3-x^2+39x-41+40-40x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}-x^3-x^2-x-1\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{5-x}\right) $ by each term in $ \left( 1-x\right) $. $$ \left( \color{blue}{5-x}\right) \cdot \left( 1-x\right) = 5-5x-x+x^2 $$Multiply $ \color{blue}{2} $ by $ \left( 7+x\right) $ $$ \color{blue}{2} \cdot \left( 7+x\right) = 14+2x $$Multiply $ \color{blue}{40} $ by $ \left( 1-x\right) $ $$ \color{blue}{40} \cdot \left( 1-x\right) = 40-40x $$ |
| ② | Combine like terms: $$ 5 \color{blue}{-5x} \color{blue}{-x} +x^2 = x^2 \color{blue}{-6x} +5 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-6x+5}\right) $ by each term in $ \left( -7-x\right) $. $$ \left( \color{blue}{x^2-6x+5}\right) \cdot \left( -7-x\right) = -7x^2-x^3+42x+6x^2-35-5x $$ |
| ④ | Combine like terms: $$ \color{blue}{-7x^2} -x^3+ \color{red}{42x} + \color{blue}{6x^2} -35 \color{red}{-5x} = -x^3 \color{blue}{-x^2} + \color{red}{37x} -35 $$ |
| ⑤ | Combine like terms: $$ -x^3-x^2+ \color{blue}{37x} \color{red}{-35} + \color{red}{14} + \color{blue}{2x} = -x^3-x^2+ \color{blue}{39x} \color{red}{-21} $$ |
| ⑥ | Combine like terms: $$ -x^3-x^2+39x \color{blue}{-21} \color{blue}{-20} = -x^3-x^2+39x \color{blue}{-41} $$ |
| ⑦ | Combine like terms: $$ -x^3-x^2+ \color{blue}{39x} \color{red}{-41} + \color{red}{40} \color{blue}{-40x} = -x^3-x^2 \color{blue}{-x} \color{red}{-1} $$ |
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