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