Question
Answer
$$5x^2(m-n)+10x^3(m-n)^3=10m^3x^3-30m^2nx^3+30mn^2x^3-10n^3x^3+5mx^2-5nx^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}5x^2(m-n)+10x^3(m-n)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}5x^2(m-n)+10x^3(1m^3-3m^2n+3mn^2-n^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}5mx^2-5nx^2+10m^3x^3-30m^2nx^3+30mn^2x^3-10n^3x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}10m^3x^3-30m^2nx^3+30mn^2x^3-10n^3x^3+5mx^2-5nx^2\end{aligned} $$ | |
| ① | Find $ \left(m-n\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = m $ and $ B = n $. $$ \left(m-n\right)^3 = m^3-3 \cdot m^2 \cdot n + 3 \cdot m \cdot n^2-n^3 = m^3-3m^2n+3mn^2-n^3 $$ |
| ② | Multiply $ \color{blue}{5x^2} $ by $ \left( m-n\right) $ $$ \color{blue}{5x^2} \cdot \left( m-n\right) = 5mx^2-5nx^2 $$Multiply $ \color{blue}{10x^3} $ by $ \left( m^3-3m^2n+3mn^2-n^3\right) $ $$ \color{blue}{10x^3} \cdot \left( m^3-3m^2n+3mn^2-n^3\right) = 10m^3x^3-30m^2nx^3+30mn^2x^3-10n^3x^3 $$ |
| ③ | Combine like terms: $$ 5mx^2-5nx^2+10m^3x^3-30m^2nx^3+30mn^2x^3-10n^3x^3 = 10m^3x^3-30m^2nx^3+30mn^2x^3-10n^3x^3+5mx^2-5nx^2 $$ |
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