Question
Answer
$$(2+x)^3+(2-x)^3+x^3=x^3+12x^2+16$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(2+x)^3+(2-x)^3+x^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}8+12x+6x^2+x^3+8-12x+6x^2-x^3+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}12x^2+16+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^3+12x^2+16\end{aligned} $$ | |
| ① | Find $ \left(2+x\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 2 $ and $ B = x $. $$ \left(2+x\right)^3 = 2^3+3 \cdot 2^2 \cdot x + 3 \cdot 2 \cdot x^2+x^3 = 8+12x+6x^2+x^3 $$Find $ \left(2-x\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 2 $ and $ B = x $. $$ \left(2-x\right)^3 = 2^3-3 \cdot 2^2 \cdot x + 3 \cdot 2 \cdot x^2-x^3 = 8-12x+6x^2-x^3 $$ |
| ② | Combine like terms: $$ \color{blue}{8} + \, \color{red}{ \cancel{12x}} \,+ \color{orange}{6x^2} + \, \color{blue}{ \cancel{x^3}} \,+ \color{blue}{8} \, \color{red}{ -\cancel{12x}} \,+ \color{orange}{6x^2} \, \color{blue}{ -\cancel{x^3}} \, = \color{orange}{12x^2} + \color{blue}{16} $$ |
| ③ | Combine like terms: $$ x^3+12x^2+16 = x^3+12x^2+16 $$ |
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