Question
Answer
$$(x^2+16)(x+2)(x-3)(2x-1)=2x^5-3x^4+21x^3-42x^2-176x+96$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x^2+16)(x+2)(x-3)(2x-1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^3+2x^2+16x+32)(x-3)(2x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^4-x^3+10x^2-16x-96)(2x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}2x^5-3x^4+21x^3-42x^2-176x+96\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x^2+16}\right) $ by each term in $ \left( x+2\right) $. $$ \left( \color{blue}{x^2+16}\right) \cdot \left( x+2\right) = x^3+2x^2+16x+32 $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^3+2x^2+16x+32}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{x^3+2x^2+16x+32}\right) \cdot \left( x-3\right) = x^4-3x^3+2x^3-6x^2+16x^2-48x+32x-96 $$ |
| ③ | Combine like terms: $$ x^4 \color{blue}{-3x^3} + \color{blue}{2x^3} \color{red}{-6x^2} + \color{red}{16x^2} \color{green}{-48x} + \color{green}{32x} -96 = x^4 \color{blue}{-x^3} + \color{red}{10x^2} \color{green}{-16x} -96 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^4-x^3+10x^2-16x-96}\right) $ by each term in $ \left( 2x-1\right) $. $$ \left( \color{blue}{x^4-x^3+10x^2-16x-96}\right) \cdot \left( 2x-1\right) = 2x^5-x^4-2x^4+x^3+20x^3-10x^2-32x^2+16x-192x+96 $$ |
| ⑤ | Combine like terms: $$ 2x^5 \color{blue}{-x^4} \color{blue}{-2x^4} + \color{red}{x^3} + \color{red}{20x^3} \color{green}{-10x^2} \color{green}{-32x^2} + \color{orange}{16x} \color{orange}{-192x} +96 = \\ = 2x^5 \color{blue}{-3x^4} + \color{red}{21x^3} \color{green}{-42x^2} \color{orange}{-176x} +96 $$ |
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