Question
Answer
$$((x-1)^3+9(x-1)^2+4(x-1)+2)\cdot2=2x^3+12x^2-22x+12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}((x-1)^3+9(x-1)^2+4(x-1)+2)\cdot2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^3-3x^2+3x-1+9(x^2-2x+1)+4(x-1)+2)\cdot2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-3x^2+3x-1+9x^2-18x+9+4x-4+2)\cdot2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3+6x^2-15x+8+4x-4+2)\cdot2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^3+6x^2-11x+4+2)\cdot2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(x^3+6x^2-11x+6)\cdot2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}2x^3+12x^2-22x+12\end{aligned} $$ | |
| 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}{9} $ by $ \left( x^2-2x+1\right) $ $$ \color{blue}{9} \cdot \left( x^2-2x+1\right) = 9x^2-18x+9 $$Multiply $ \color{blue}{4} $ by $ \left( x-1\right) $ $$ \color{blue}{4} \cdot \left( x-1\right) = 4x-4 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-3x^2} + \color{red}{3x} \color{green}{-1} + \color{blue}{9x^2} \color{red}{-18x} + \color{green}{9} = x^3+ \color{blue}{6x^2} \color{red}{-15x} + \color{green}{8} $$ |
| ④ | Combine like terms: $$ x^3+6x^2 \color{blue}{-15x} + \color{red}{8} + \color{blue}{4x} \color{red}{-4} = x^3+6x^2 \color{blue}{-11x} + \color{red}{4} $$ |
| ⑤ | Combine like terms: $$ x^3+6x^2-11x+ \color{blue}{4} + \color{blue}{2} = x^3+6x^2-11x+ \color{blue}{6} $$ |
| ⑥ | $$ \left( \color{blue}{x^3+6x^2-11x+6}\right) \cdot 2 = 2x^3+12x^2-22x+12 $$ |
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