Question
Answer
$$(x+1)^3+5(x+1)^2-(x+1)+7=x^3+8x^2+12x+12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+1)^3+5(x+1)^2-(x+1)+7& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^3+3x^2+3x+1+5(x^2+2x+1)-(x+1)+7 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^3+3x^2+3x+1+5x^2+10x+5-(x+1)+7 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^3+8x^2+13x+6-(x+1)+7 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^3+8x^2+13x+6-x-1+7 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^3+8x^2+12x+12\end{aligned} $$ | |
| 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(x+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}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$ |
| ② | Multiply $ \color{blue}{5} $ by $ \left( x^2+2x+1\right) $ $$ \color{blue}{5} \cdot \left( x^2+2x+1\right) = 5x^2+10x+5 $$ |
| ③ | Combine like terms: $$ x^3+ \color{blue}{3x^2} + \color{red}{3x} + \color{green}{1} + \color{blue}{5x^2} + \color{red}{10x} + \color{green}{5} = x^3+ \color{blue}{8x^2} + \color{red}{13x} + \color{green}{6} $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( x+1 \right) = -x-1 $$ |
| ⑤ | Combine like terms: $$ x^3+8x^2+ \color{blue}{13x} + \color{red}{6} \color{blue}{-x} \color{green}{-1} + \color{green}{7} = x^3+8x^2+ \color{blue}{12x} + \color{green}{12} $$ |
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