Question
Answer
$$(2+h)^3-(2+h)^2+8=h^3+5h^2+8h+12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(2+h)^3-(2+h)^2+8& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}8+12h+6h^2+h^3-(4+4h+h^2)+8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}8+12h+6h^2+h^3-4-4h-h^2+8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}h^3+5h^2+8h+12\end{aligned} $$ | |
| ① | Find $ \left(2+h\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 2 $ and $ B = h $. $$ \left(2+h\right)^3 = 2^3+3 \cdot 2^2 \cdot h + 3 \cdot 2 \cdot h^2+h^3 = 8+12h+6h^2+h^3 $$Find $ \left(2+h\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 2 } $ and $ B = \color{red}{ h }$. $$ \begin{aligned}\left(2+h\right)^2 = \color{blue}{2^2} +2 \cdot 2 \cdot h + \color{red}{h^2} = 4+4h+h^2\end{aligned} $$ |
| ② | Remove the parentheses by changing the sign of each term within them. $$ - \left( 4+4h+h^2 \right) = -4-4h-h^2 $$ |
| ③ | Combine like terms: $$ \color{blue}{8} + \color{red}{12h} + \color{green}{6h^2} +h^3 \color{orange}{-4} \color{red}{-4h} \color{green}{-h^2} + \color{orange}{8} = h^3+ \color{green}{5h^2} + \color{red}{8h} + \color{orange}{12} $$ |
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