Question
Answer
$$4(u+x)^4+4(v+x)^4=4u^4+16u^3x+24u^2x^2+16ux^3+4v^4+16v^3x+24v^2x^2+16vx^3+8x^4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}4(u+x)^4+4(v+x)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}4(1u^4+4u^3x+6u^2x^2+4ux^3+x^4)+4(1v^4+4v^3x+6v^2x^2+4vx^3+x^4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}4u^4+16u^3x+24u^2x^2+16ux^3+4x^4+4v^4+16v^3x+24v^2x^2+16vx^3+4x^4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}4u^4+16u^3x+24u^2x^2+16ux^3+4v^4+16v^3x+24v^2x^2+16vx^3+8x^4\end{aligned} $$ | |
| ① | $$ (u+x)^4 = (u+x)^2 \cdot (u+x)^2 $$ |
| ② | Find $ \left(u+x\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ u } $ and $ B = \color{red}{ x }$. $$ \begin{aligned}\left(u+x\right)^2 = \color{blue}{u^2} +2 \cdot u \cdot x + \color{red}{x^2} = u^2+2ux+x^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{u^2+2ux+x^2}\right) $ by each term in $ \left( u^2+2ux+x^2\right) $. $$ \left( \color{blue}{u^2+2ux+x^2}\right) \cdot \left( u^2+2ux+x^2\right) = \\ = u^4+2u^3x+u^2x^2+2u^3x+4u^2x^2+2ux^3+u^2x^2+2ux^3+x^4 $$ |
| ④ | Combine like terms: $$ u^4+ \color{blue}{2u^3x} + \color{red}{u^2x^2} + \color{blue}{2u^3x} + \color{green}{4u^2x^2} + \color{orange}{2ux^3} + \color{green}{u^2x^2} + \color{orange}{2ux^3} +x^4 = \\ = u^4+ \color{blue}{4u^3x} + \color{green}{6u^2x^2} + \color{orange}{4ux^3} +x^4 $$$$ (v+x)^4 = (v+x)^2 \cdot (v+x)^2 $$ |
| ⑤ | Find $ \left(v+x\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ v } $ and $ B = \color{red}{ x }$. $$ \begin{aligned}\left(v+x\right)^2 = \color{blue}{v^2} +2 \cdot v \cdot x + \color{red}{x^2} = v^2+2vx+x^2\end{aligned} $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{v^2+2vx+x^2}\right) $ by each term in $ \left( v^2+2vx+x^2\right) $. $$ \left( \color{blue}{v^2+2vx+x^2}\right) \cdot \left( v^2+2vx+x^2\right) = \\ = v^4+2v^3x+v^2x^2+2v^3x+4v^2x^2+2vx^3+v^2x^2+2vx^3+x^4 $$ |
| ⑦ | Combine like terms: $$ v^4+ \color{blue}{2v^3x} + \color{red}{v^2x^2} + \color{blue}{2v^3x} + \color{green}{4v^2x^2} + \color{orange}{2vx^3} + \color{green}{v^2x^2} + \color{orange}{2vx^3} +x^4 = \\ = v^4+ \color{blue}{4v^3x} + \color{green}{6v^2x^2} + \color{orange}{4vx^3} +x^4 $$ |
| ⑧ | Multiply $ \color{blue}{4} $ by $ \left( u^4+4u^3x+6u^2x^2+4ux^3+x^4\right) $ $$ \color{blue}{4} \cdot \left( u^4+4u^3x+6u^2x^2+4ux^3+x^4\right) = 4u^4+16u^3x+24u^2x^2+16ux^3+4x^4 $$Multiply $ \color{blue}{4} $ by $ \left( v^4+4v^3x+6v^2x^2+4vx^3+x^4\right) $ $$ \color{blue}{4} \cdot \left( v^4+4v^3x+6v^2x^2+4vx^3+x^4\right) = 4v^4+16v^3x+24v^2x^2+16vx^3+4x^4 $$ |
| ⑨ | Combine like terms: $$ 4u^4+16u^3x+24u^2x^2+16ux^3+ \color{blue}{4x^4} +4v^4+16v^3x+24v^2x^2+16vx^3+ \color{blue}{4x^4} = \\ = 4u^4+16u^3x+24u^2x^2+16ux^3+4v^4+16v^3x+24v^2x^2+16vx^3+ \color{blue}{8x^4} $$ |
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