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