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