Question
Answer
$$2(x+1)^3+3(k+1)^2+k+1=2x^3+3k^2+6x^2+7k+6x+6$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2(x+1)^3+3(k+1)^2+k+1& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}2(x^3+3x^2+3x+1)+3(1k^2+2k+1)+k+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}2x^3+6x^2+6x+2+3k^2+6k+3+k+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}2x^3+3k^2+6x^2+6k+6x+5+k+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2x^3+3k^2+6x^2+7k+6x+6\end{aligned} $$ | |
| ① | Find $ \left(x+1\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 1 $. $$ \left(x+1\right)^3 = x^3+3 \cdot x^2 \cdot 1 + 3 \cdot x \cdot 1^2+1^3 = x^3+3x^2+3x+1 $$Find $ \left(k+1\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ k } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(k+1\right)^2 = \color{blue}{k^2} +2 \cdot k \cdot 1 + \color{red}{1^2} = k^2+2k+1\end{aligned} $$ |
| ② | Multiply $ \color{blue}{2} $ by $ \left( x^3+3x^2+3x+1\right) $ $$ \color{blue}{2} \cdot \left( x^3+3x^2+3x+1\right) = 2x^3+6x^2+6x+2 $$Multiply $ \color{blue}{3} $ by $ \left( k^2+2k+1\right) $ $$ \color{blue}{3} \cdot \left( k^2+2k+1\right) = 3k^2+6k+3 $$ |
| ③ | Combine like terms: $$ 2x^3+6x^2+6x+ \color{blue}{2} +3k^2+6k+ \color{blue}{3} = 2x^3+3k^2+6x^2+6k+6x+ \color{blue}{5} $$ |
| ④ | Combine like terms: $$ 2x^3+3k^2+6x^2+ \color{blue}{6k} +6x+ \color{red}{5} + \color{blue}{k} + \color{red}{1} = 2x^3+3k^2+6x^2+ \color{blue}{7k} +6x+ \color{red}{6} $$ |
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