Question
Answer
$$(x+10)^3(x+2)(x-3)=x^5+29x^4+264x^3+520x^2-2800x-6000$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+10)^3(x+2)(x-3)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^3+30x^2+300x+1000)(x+2)(x-3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^4+32x^3+360x^2+1600x+2000)(x-3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^5+29x^4+264x^3+520x^2-2800x-6000\end{aligned} $$ | |
| ① | Find $ \left(x+10\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 10 $. $$ \left(x+10\right)^3 = x^3+3 \cdot x^2 \cdot 10 + 3 \cdot x \cdot 10^2+10^3 = x^3+30x^2+300x+1000 $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^3+30x^2+300x+1000}\right) $ by each term in $ \left( x+2\right) $. $$ \left( \color{blue}{x^3+30x^2+300x+1000}\right) \cdot \left( x+2\right) = x^4+2x^3+30x^3+60x^2+300x^2+600x+1000x+2000 $$ |
| ③ | Combine like terms: $$ x^4+ \color{blue}{2x^3} + \color{blue}{30x^3} + \color{red}{60x^2} + \color{red}{300x^2} + \color{green}{600x} + \color{green}{1000x} +2000 = \\ = x^4+ \color{blue}{32x^3} + \color{red}{360x^2} + \color{green}{1600x} +2000 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^4+32x^3+360x^2+1600x+2000}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{x^4+32x^3+360x^2+1600x+2000}\right) \cdot \left( x-3\right) = \\ = x^5-3x^4+32x^4-96x^3+360x^3-1080x^2+1600x^2-4800x+2000x-6000 $$ |
| ⑤ | Combine like terms: $$ x^5 \color{blue}{-3x^4} + \color{blue}{32x^4} \color{red}{-96x^3} + \color{red}{360x^3} \color{green}{-1080x^2} + \color{green}{1600x^2} \color{orange}{-4800x} + \color{orange}{2000x} -6000 = \\ = x^5+ \color{blue}{29x^4} + \color{red}{264x^3} + \color{green}{520x^2} \color{orange}{-2800x} -6000 $$ |
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