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