Question
Answer
$$-2x(x-5)^3(x+1)^2=-2x^6+26x^5-92x^4-20x^3+350x^2+250x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-2x(x-5)^3(x+1)^2& \xlongequal{ }-2x(x^3-15x^2+75x-125)(x^2+2x+1) \xlongequal{ } \\[1 em] & \xlongequal{ }-(2x^4-30x^3+150x^2-250x)(x^2+2x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}-(2x^6-26x^5+92x^4+20x^3-350x^2-250x) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-2x^6+26x^5-92x^4-20x^3+350x^2+250x\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{2x^4-30x^3+150x^2-250x}\right) $ by each term in $ \left( x^2+2x+1\right) $. $$ \left( \color{blue}{2x^4-30x^3+150x^2-250x}\right) \cdot \left( x^2+2x+1\right) = \\ = 2x^6+4x^5+2x^4-30x^5-60x^4-30x^3+150x^4+300x^3+150x^2-250x^3-500x^2-250x $$ |
| ② | Combine like terms: $$ 2x^6+ \color{blue}{4x^5} + \color{red}{2x^4} \color{blue}{-30x^5} \color{green}{-60x^4} \color{orange}{-30x^3} + \color{green}{150x^4} + \color{blue}{300x^3} + \color{red}{150x^2} \color{blue}{-250x^3} \color{red}{-500x^2} -250x = \\ = 2x^6 \color{blue}{-26x^5} + \color{green}{92x^4} + \color{blue}{20x^3} \color{red}{-350x^2} -250x $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left(2x^6-26x^5+92x^4+20x^3-350x^2-250x \right) = -2x^6+26x^5-92x^4-20x^3+350x^2+250x $$ |
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