Question
Answer
$$4(x^2+1)(x-2)^3=4x^5-24x^4+52x^3-56x^2+48x-32$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}4(x^2+1)(x-2)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}4(x^2+1)(x^3-6x^2+12x-8) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(4x^2+4)(x^3-6x^2+12x-8) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}4x^5-24x^4+52x^3-56x^2+48x-32\end{aligned} $$ | |
| 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 $. $$ \left(x-2\right)^3 = x^3-3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2-2^3 = x^3-6x^2+12x-8 $$ |
| ② | Multiply $ \color{blue}{4} $ by $ \left( x^2+1\right) $ $$ \color{blue}{4} \cdot \left( x^2+1\right) = 4x^2+4 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{4x^2+4}\right) $ by each term in $ \left( x^3-6x^2+12x-8\right) $. $$ \left( \color{blue}{4x^2+4}\right) \cdot \left( x^3-6x^2+12x-8\right) = 4x^5-24x^4+48x^3-32x^2+4x^3-24x^2+48x-32 $$ |
| ④ | Combine like terms: $$ 4x^5-24x^4+ \color{blue}{48x^3} \color{red}{-32x^2} + \color{blue}{4x^3} \color{red}{-24x^2} +48x-32 = 4x^5-24x^4+ \color{blue}{52x^3} \color{red}{-56x^2} +48x-32 $$ |
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