Question
Answer
$$2x(x-2)^3+3(x-2)^2x^2=5x^4-24x^3+36x^2-16x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2x(x-2)^3+3(x-2)^2x^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}2x(x^3-6x^2+12x-8)+3(x^2-4x+4)x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}2x^4-12x^3+24x^2-16x+(3x^2-12x+12)x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}2x^4-12x^3+24x^2-16x+3x^4-12x^3+12x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}5x^4-24x^3+36x^2-16x\end{aligned} $$ | |
| 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}{2x} $ by $ \left( x^3-6x^2+12x-8\right) $ $$ \color{blue}{2x} \cdot \left( x^3-6x^2+12x-8\right) = 2x^4-12x^3+24x^2-16x $$Multiply $ \color{blue}{3} $ by $ \left( x^2-4x+4\right) $ $$ \color{blue}{3} \cdot \left( x^2-4x+4\right) = 3x^2-12x+12 $$ |
| ③ | $$ \left( \color{blue}{3x^2-12x+12}\right) \cdot x^2 = 3x^4-12x^3+12x^2 $$ |
| ④ | Combine like terms: $$ \color{blue}{2x^4} \color{red}{-12x^3} + \color{green}{24x^2} -16x+ \color{blue}{3x^4} \color{red}{-12x^3} + \color{green}{12x^2} = \color{blue}{5x^4} \color{red}{-24x^3} + \color{green}{36x^2} -16x $$ |
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