Question
Answer
$$-a^2(b-a)^2-2ab(b^2-a^2)-2a(b^2-a^2)=-a^4+4a^3b-a^2b^2-2ab^3+2a^3-2ab^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-a^2(b-a)^2-2ab(b^2-a^2)-2a(b^2-a^2)& \xlongequal{ }-a^2(1b^2-2ab+a^2)-2ab(b^2-a^2)-2a(b^2-a^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}-(1a^2b^2-2a^3b+a^4)-(2ab^3-2a^3b)-(2ab^2-2a^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}-a^2b^2+2a^3b-a^4-(2ab^3-2a^3b)-(2ab^2-2a^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-a^2b^2+2a^3b-a^4-2ab^3+2a^3b-(2ab^2-2a^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-a^4+4a^3b-a^2b^2-2ab^3-(2ab^2-2a^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-a^4+4a^3b-a^2b^2-2ab^3-2ab^2+2a^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-a^4+4a^3b-a^2b^2-2ab^3+2a^3-2ab^2\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{a^2} $ by $ \left( b^2-2ab+a^2\right) $ $$ \color{blue}{a^2} \cdot \left( b^2-2ab+a^2\right) = a^2b^2-2a^3b+a^4 $$Multiply $ \color{blue}{2ab} $ by $ \left( b^2-a^2\right) $ $$ \color{blue}{2ab} \cdot \left( b^2-a^2\right) = 2ab^3-2a^3b $$Multiply $ \color{blue}{2a} $ by $ \left( b^2-a^2\right) $ $$ \color{blue}{2a} \cdot \left( b^2-a^2\right) = 2ab^2-2a^3 $$ |
| ② | Remove the parentheses by changing the sign of each term within them. $$ - \left(a^2b^2-2a^3b+a^4 \right) = -a^2b^2+2a^3b-a^4 $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 2ab^3-2a^3b \right) = -2ab^3+2a^3b $$ |
| ④ | Combine like terms: $$ -a^2b^2+ \color{blue}{2a^3b} -a^4-2ab^3+ \color{blue}{2a^3b} = -a^4+ \color{blue}{4a^3b} -a^2b^2-2ab^3 $$ |
| ⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 2ab^2-2a^3 \right) = -2ab^2+2a^3 $$ |
| ⑥ | Combine like terms: $$ -a^4+4a^3b-a^2b^2-2ab^3+2a^3-2ab^2 = -a^4+4a^3b-a^2b^2-2ab^3+2a^3-2ab^2 $$ |
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