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