Question
Answer
$$(x+2h)^3-2(x+h)^3+x^3=6h^3+6h^2x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+2h)^3-2(x+h)^3+x^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^3+6hx^2+12h^2x+8h^3-2(x^3+3hx^2+3h^2x+h^3)+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^3+6hx^2+12h^2x+8h^3-(2x^3+6hx^2+6h^2x+2h^3)+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^3+6hx^2+12h^2x+8h^3-2x^3-6hx^2-6h^2x-2h^3+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ }x^3+ \cancel{6hx^2}+12h^2x+8h^3-2x^3 -\cancel{6hx^2}-6h^2x-2h^3+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}6h^3+6h^2x\end{aligned} $$ | |
| ① | Find $ \left(x+2h\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 2h $. $$ \left(x+2h\right)^3 = x^3+3 \cdot x^2 \cdot 2h + 3 \cdot x \cdot \left( 2h \right)^2+\left( 2h \right)^3 = x^3+6hx^2+12h^2x+8h^3 $$Find $ \left(x+h\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = h $. $$ \left(x+h\right)^3 = x^3+3 \cdot x^2 \cdot h + 3 \cdot x \cdot h^2+h^3 = x^3+3hx^2+3h^2x+h^3 $$ |
| ② | Multiply $ \color{blue}{2} $ by $ \left( x^3+3hx^2+3h^2x+h^3\right) $ $$ \color{blue}{2} \cdot \left( x^3+3hx^2+3h^2x+h^3\right) = 2x^3+6hx^2+6h^2x+2h^3 $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 2x^3+6hx^2+6h^2x+2h^3 \right) = -2x^3-6hx^2-6h^2x-2h^3 $$ |
| ④ | Combine like terms: $$ \color{blue}{x^3} + \, \color{red}{ \cancel{6hx^2}} \,+ \color{orange}{12h^2x} + \color{blue}{8h^3} \color{red}{-2x^3} \, \color{red}{ -\cancel{6hx^2}} \, \color{orange}{-6h^2x} \color{blue}{-2h^3} + \color{red}{x^3} = \color{blue}{6h^3} + \color{orange}{6h^2x} $$ |
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