Question
Answer
$$(1-x)\cdot(1-x^5)-(x-x^{20})\cdot(1-x^5)-x^{24}\cdot(1-x)=-x^{24}+x^{20}+2x^6-x^5-2x+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1-x)\cdot(1-x^5)-(x-x^{20})\cdot(1-x^5)-x^{24}\cdot(1-x)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}1-x^5-x+x^6-(x-x^6-x^{20}+x^{25})-(x^{24}-x^{25}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}1-x^5-x+x^6-x+x^6+x^{20}-x^{25}-(x^{24}-x^{25}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-x^{25}+x^{20}+2x^6-x^5-2x+1-(x^{24}-x^{25}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-x^{25}+x^{20}+2x^6-x^5-2x+1-x^{24}+x^{25} \xlongequal{ } \\[1 em] & \xlongequal{ } -\cancel{x^{25}}+x^{20}+2x^6-x^5-2x+1-x^{24}+ \cancel{x^{25}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-x^{24}+x^{20}+2x^6-x^5-2x+1\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1-x}\right) $ by each term in $ \left( 1-x^5\right) $. $$ \left( \color{blue}{1-x}\right) \cdot \left( 1-x^5\right) = 1-x^5-x+x^6 $$Multiply each term of $ \left( \color{blue}{x-x^{20}}\right) $ by each term in $ \left( 1-x^5\right) $. $$ \left( \color{blue}{x-x^{20}}\right) \cdot \left( 1-x^5\right) = x-x^6-x^{20}+x^{25} $$Multiply $ \color{blue}{x^{24}} $ by $ \left( 1-x\right) $ $$ \color{blue}{x^{24}} \cdot \left( 1-x\right) = x^{24}-x^{25} $$ |
| ② | Remove the parentheses by changing the sign of each term within them. $$ - \left( x-x^6-x^{20}+x^{25} \right) = -x+x^6+x^{20}-x^{25} $$ |
| ③ | Combine like terms: $$ 1-x^5 \color{blue}{-x} + \color{red}{x^6} \color{blue}{-x} + \color{red}{x^6} +x^{20}-x^{25} = -x^{25}+x^{20}+ \color{red}{2x^6} -x^5 \color{blue}{-2x} +1 $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( x^{24}-x^{25} \right) = -x^{24}+x^{25} $$ |
| ⑤ | Combine like terms: $$ \, \color{blue}{ -\cancel{x^{25}}} \,+x^{20}+2x^6-x^5-2x+1-x^{24}+ \, \color{blue}{ \cancel{x^{25}}} \, = -x^{24}+x^{20}+2x^6-x^5-2x+1 $$ |
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