Question
Answer
$$(x-2)^3+9(x-2)^2+4(x-2)+2=x^3+3x^2-20x+22$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-2)^3+9(x-2)^2+4(x-2)+2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^3-6x^2+12x-8+9(x^2-4x+4)+4(x-2)+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^3-6x^2+12x-8+9x^2-36x+36+4x-8+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^3+3x^2-24x+28+4x-8+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^3+3x^2-20x+20+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^3+3x^2-20x+22\end{aligned} $$ | |
| ① | Find $ \left(x-2\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = x $ and $ B = 2 $. $$ \left(x-2\right)^3 = x^3-3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2-2^3 = x^3-6x^2+12x-8 $$Find $ \left(x-2\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}{ 2 }$. $$ \begin{aligned}\left(x-2\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 2 + \color{red}{2^2} = x^2-4x+4\end{aligned} $$ |
| ② | Multiply $ \color{blue}{9} $ by $ \left( x^2-4x+4\right) $ $$ \color{blue}{9} \cdot \left( x^2-4x+4\right) = 9x^2-36x+36 $$Multiply $ \color{blue}{4} $ by $ \left( x-2\right) $ $$ \color{blue}{4} \cdot \left( x-2\right) = 4x-8 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-6x^2} + \color{red}{12x} \color{green}{-8} + \color{blue}{9x^2} \color{red}{-36x} + \color{green}{36} = x^3+ \color{blue}{3x^2} \color{red}{-24x} + \color{green}{28} $$ |
| ④ | Combine like terms: $$ x^3+3x^2 \color{blue}{-24x} + \color{red}{28} + \color{blue}{4x} \color{red}{-8} = x^3+3x^2 \color{blue}{-20x} + \color{red}{20} $$ |
| ⑤ | Combine like terms: $$ x^3+3x^2-20x+ \color{blue}{20} + \color{blue}{2} = x^3+3x^2-20x+ \color{blue}{22} $$ |
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