Question
Answer
$$2x(x-3)^3(x-2)=2x^5-22x^4+90x^3-162x^2+108x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2x(x-3)^3(x-2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}2x(x^3-9x^2+27x-27)(x-2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(2x^4-18x^3+54x^2-54x)(x-2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2x^5-22x^4+90x^3-162x^2+108x\end{aligned} $$ | |
| ① | Find $ \left(x-3\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = x $ and $ B = 3 $. $$ \left(x-3\right)^3 = x^3-3 \cdot x^2 \cdot 3 + 3 \cdot x \cdot 3^2-3^3 = x^3-9x^2+27x-27 $$ |
| ② | Multiply $ \color{blue}{2x} $ by $ \left( x^3-9x^2+27x-27\right) $ $$ \color{blue}{2x} \cdot \left( x^3-9x^2+27x-27\right) = 2x^4-18x^3+54x^2-54x $$ |
| ③ | Multiply each term of $ \left( \color{blue}{2x^4-18x^3+54x^2-54x}\right) $ by each term in $ \left( x-2\right) $. $$ \left( \color{blue}{2x^4-18x^3+54x^2-54x}\right) \cdot \left( x-2\right) = 2x^5-4x^4-18x^4+36x^3+54x^3-108x^2-54x^2+108x $$ |
| ④ | Combine like terms: $$ 2x^5 \color{blue}{-4x^4} \color{blue}{-18x^4} + \color{red}{36x^3} + \color{red}{54x^3} \color{green}{-108x^2} \color{green}{-54x^2} +108x = \\ = 2x^5 \color{blue}{-22x^4} + \color{red}{90x^3} \color{green}{-162x^2} +108x $$ |
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