Question
Answer
$$x^2(x^2+3y^2+z^2)^2+4y^2(y^2+z^2)^2=x^6+6x^4y^2+2x^4z^2+9x^2y^4+6x^2y^2z^2+x^2z^4+4y^6+8y^4z^2+4y^2z^4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x^2(x^2+3y^2+z^2)^2+4y^2(y^2+z^2)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^2(x^4+6x^2y^2+2x^2z^2+9y^4+6y^2z^2+z^4)+4y^2(1y^4+2y^2z^2+z^4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^6+6x^4y^2+2x^4z^2+9x^2y^4+6x^2y^2z^2+x^2z^4+4y^6+8y^4z^2+4y^2z^4\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x^2+3y^2+z^2}\right) $ by each term in $ \left( x^2+3y^2+z^2\right) $. $$ \left( \color{blue}{x^2+3y^2+z^2}\right) \cdot \left( x^2+3y^2+z^2\right) = \\ = x^4+3x^2y^2+x^2z^2+3x^2y^2+9y^4+3y^2z^2+x^2z^2+3y^2z^2+z^4 $$ |
| ② | Combine like terms: $$ x^4+ \color{blue}{3x^2y^2} + \color{red}{x^2z^2} + \color{blue}{3x^2y^2} +9y^4+ \color{green}{3y^2z^2} + \color{red}{x^2z^2} + \color{green}{3y^2z^2} +z^4 = \\ = x^4+ \color{blue}{6x^2y^2} + \color{red}{2x^2z^2} +9y^4+ \color{green}{6y^2z^2} +z^4 $$Find $ \left(y^2+z^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}{ y^2 } $ and $ B = \color{red}{ z^2 }$. $$ \begin{aligned}\left(y^2+z^2\right)^2 = \color{blue}{\left( y^2 \right)^2} +2 \cdot y^2 \cdot z^2 + \color{red}{\left( z^2 \right)^2} = y^4+2y^2z^2+z^4\end{aligned} $$ |
| ③ | Multiply $ \color{blue}{x^2} $ by $ \left( x^4+6x^2y^2+2x^2z^2+9y^4+6y^2z^2+z^4\right) $ $$ \color{blue}{x^2} \cdot \left( x^4+6x^2y^2+2x^2z^2+9y^4+6y^2z^2+z^4\right) = x^6+6x^4y^2+2x^4z^2+9x^2y^4+6x^2y^2z^2+x^2z^4 $$Multiply $ \color{blue}{4y^2} $ by $ \left( y^4+2y^2z^2+z^4\right) $ $$ \color{blue}{4y^2} \cdot \left( y^4+2y^2z^2+z^4\right) = 4y^6+8y^4z^2+4y^2z^4 $$ |
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