Question
Answer
$$x(x+x(3x-2x^2+x^3)-x^2(3x-2x^2+x^3))(2x-x^2)=x^8-5x^7+11x^6-13x^5+5x^4+2x^3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x(x+x(3x-2x^2+x^3)-x^2(3x-2x^2+x^3))(2x-x^2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x(x+3x^2-2x^3+x^4-(3x^3-2x^4+x^5))(2x-x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x(x^4-2x^3+3x^2+x-(3x^3-2x^4+x^5))(2x-x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x(x^4-2x^3+3x^2+x-3x^3+2x^4-x^5)(2x-x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x(-x^5+3x^4-5x^3+3x^2+x)(2x-x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(-x^6+3x^5-5x^4+3x^3+x^2)(2x-x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}x^8-5x^7+11x^6-13x^5+5x^4+2x^3\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{x} $ by $ \left( 3x-2x^2+x^3\right) $ $$ \color{blue}{x} \cdot \left( 3x-2x^2+x^3\right) = 3x^2-2x^3+x^4 $$Multiply $ \color{blue}{x^2} $ by $ \left( 3x-2x^2+x^3\right) $ $$ \color{blue}{x^2} \cdot \left( 3x-2x^2+x^3\right) = 3x^3-2x^4+x^5 $$ |
| ② | Combine like terms: $$ x+3x^2-2x^3+x^4 = x^4-2x^3+3x^2+x $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 3x^3-2x^4+x^5 \right) = -3x^3+2x^4-x^5 $$ |
| ④ | Combine like terms: $$ \color{blue}{x^4} \color{red}{-2x^3} +3x^2+x \color{red}{-3x^3} + \color{blue}{2x^4} -x^5 = -x^5+ \color{blue}{3x^4} \color{red}{-5x^3} +3x^2+x $$ |
| ⑤ | Multiply $ \color{blue}{x} $ by $ \left( -x^5+3x^4-5x^3+3x^2+x\right) $ $$ \color{blue}{x} \cdot \left( -x^5+3x^4-5x^3+3x^2+x\right) = -x^6+3x^5-5x^4+3x^3+x^2 $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{-x^6+3x^5-5x^4+3x^3+x^2}\right) $ by each term in $ \left( 2x-x^2\right) $. $$ \left( \color{blue}{-x^6+3x^5-5x^4+3x^3+x^2}\right) \cdot \left( 2x-x^2\right) = -2x^7+x^8+6x^6-3x^7-10x^5+5x^6+6x^4-3x^5+2x^3-x^4 $$ |
| ⑦ | Combine like terms: $$ \color{blue}{-2x^7} +x^8+ \color{red}{6x^6} \color{blue}{-3x^7} \color{green}{-10x^5} + \color{red}{5x^6} + \color{orange}{6x^4} \color{green}{-3x^5} +2x^3 \color{orange}{-x^4} = \\ = x^8 \color{blue}{-5x^7} + \color{red}{11x^6} \color{green}{-13x^5} + \color{orange}{5x^4} +2x^3 $$ |
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