Question
Answer
$$(x^2y-y^2x+y^2x-z^2y+z^2x-xz^2)^2+2(xyz)^2=x^4y^2+y^2z^4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x^2y-y^2x+y^2x-z^2y+z^2x-xz^2)^2+2(xyz)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2y-yz^2)^2+2(xyz)^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^4y^2-2x^2y^2z^2+y^2z^4+21x^2y^2z^2 \xlongequal{ } \\[1 em] & \xlongequal{ }x^4y^2-2x^2y^2z^2+y^2z^4+2x^2y^2z^2 \xlongequal{ } \\[1 em] & \xlongequal{ }x^4y^2 -\cancel{2x^2y^2z^2}+y^2z^4+ \cancel{2x^2y^2z^2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^4y^2+y^2z^4\end{aligned} $$ | |
| ① | Combine like terms: $$ x^2y \, \color{blue}{ -\cancel{xy^2}} \,+ \, \color{blue}{ \cancel{xy^2}} \,-yz^2+ \, \color{green}{ \cancel{xz^2}} \, \, \color{green}{ -\cancel{xz^2}} \, = x^2y-yz^2 $$ |
| ② | Find $ \left(x^2y-yz^2\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x^2y } $ and $ B = \color{red}{ yz^2 }$. $$ \begin{aligned}\left(x^2y-yz^2\right)^2 = \color{blue}{\left( x^2y \right)^2} -2 \cdot x^2y \cdot yz^2 + \color{red}{\left( yz^2 \right)^2} = x^4y^2-2x^2y^2z^2+y^2z^4\end{aligned} $$$$ \left( xyz \right)^2 = 1^2x^2y^2z^2 = x^2y^2z^2 $$ |
| ③ | Combine like terms: $$ x^4y^2 \, \color{blue}{ -\cancel{2x^2y^2z^2}} \,+y^2z^4+ \, \color{blue}{ \cancel{2x^2y^2z^2}} \, = x^4y^2+y^2z^4 $$ |
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