Question
Answer
$$(x-y-1)^3=x^3-3x^2y+3xy^2-y^3-3x^2+6xy-3y^2+3x-3y-1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-y-1)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^3-3x^2y+3xy^2-y^3-3x^2+6xy-3y^2+3x-3y-1\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-y-1}\right) $ by each term in $ \left( x-y-1\right) $. $$ \left( \color{blue}{x-y-1}\right) \cdot \left( x-y-1\right) = x^2-xy-x-xy+y^2+y-x+y+1 $$ |
| ② | Combine like terms: $$ x^2 \color{blue}{-xy} \color{red}{-x} \color{blue}{-xy} +y^2+ \color{green}{y} \color{red}{-x} + \color{green}{y} +1 = x^2 \color{blue}{-2xy} +y^2 \color{red}{-2x} + \color{green}{2y} +1 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-2xy+y^2-2x+2y+1}\right) $ by each term in $ \left( x-y-1\right) $. $$ \left( \color{blue}{x^2-2xy+y^2-2x+2y+1}\right) \cdot \left( x-y-1\right) = \\ = x^3-x^2y-x^2-2x^2y+2xy^2+2xy+xy^2-y^3-y^2-2x^2+2xy+2x+2xy-2y^2-2y+x-y-1 $$ |
| ④ | Combine like terms: $$ x^3 \color{blue}{-x^2y} \color{red}{-x^2} \color{blue}{-2x^2y} + \color{green}{2xy^2} + \color{orange}{2xy} + \color{green}{xy^2} -y^3 \color{blue}{-y^2} \color{red}{-2x^2} + \color{red}{2xy} + \color{green}{2x} + \color{red}{2xy} \color{blue}{-2y^2} \color{orange}{-2y} + \color{green}{x} \color{orange}{-y} -1 = \\ = x^3 \color{blue}{-3x^2y} + \color{green}{3xy^2} -y^3 \color{red}{-3x^2} + \color{red}{6xy} \color{blue}{-3y^2} + \color{green}{3x} \color{orange}{-3y} -1 $$ |
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