Question
Answer
$$(x-2)^2(x+5)^3=x^5+11x^4+19x^3-115x^2-200x+500$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-2)^2(x+5)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-4x+4)(x^3+15x^2+75x+125) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^5+11x^4+19x^3-115x^2-200x+500\end{aligned} $$ | |
| ① | 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 }$. $$ \begin{aligned}\left(x-2\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 2 + \color{red}{2^2} = x^2-4x+4\end{aligned} $$Find $ \left(x+5\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 5 $. $$ \left(x+5\right)^3 = x^3+3 \cdot x^2 \cdot 5 + 3 \cdot x \cdot 5^2+5^3 = x^3+15x^2+75x+125 $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2-4x+4}\right) $ by each term in $ \left( x^3+15x^2+75x+125\right) $. $$ \left( \color{blue}{x^2-4x+4}\right) \cdot \left( x^3+15x^2+75x+125\right) = \\ = x^5+15x^4+75x^3+125x^2-4x^4-60x^3-300x^2-500x+4x^3+60x^2+300x+500 $$ |
| ③ | Combine like terms: $$ x^5+ \color{blue}{15x^4} + \color{red}{75x^3} + \color{green}{125x^2} \color{blue}{-4x^4} \color{orange}{-60x^3} \color{blue}{-300x^2} \color{red}{-500x} + \color{orange}{4x^3} + \color{blue}{60x^2} + \color{red}{300x} +500 = \\ = x^5+ \color{blue}{11x^4} + \color{orange}{19x^3} \color{blue}{-115x^2} \color{red}{-200x} +500 $$ |
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