Question
Answer
$$(x-7)(x-2)^2(x+4)=x^4-7x^3-12x^2+100x-112$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-7)(x-2)^2(x+4)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x-7)(x^2-4x+4)(x+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-4x^2+4x-7x^2+28x-28)(x+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-11x^2+32x-28)(x+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-7x^3-12x^2+100x-112\end{aligned} $$ | |
| 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 }$. $$ \begin{aligned}\left(x-2\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 2 + \color{red}{2^2} = x^2-4x+4\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x-7}\right) $ by each term in $ \left( x^2-4x+4\right) $. $$ \left( \color{blue}{x-7}\right) \cdot \left( x^2-4x+4\right) = x^3-4x^2+4x-7x^2+28x-28 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-4x^2} + \color{red}{4x} \color{blue}{-7x^2} + \color{red}{28x} -28 = x^3 \color{blue}{-11x^2} + \color{red}{32x} -28 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-11x^2+32x-28}\right) $ by each term in $ \left( x+4\right) $. $$ \left( \color{blue}{x^3-11x^2+32x-28}\right) \cdot \left( x+4\right) = x^4+4x^3-11x^3-44x^2+32x^2+128x-28x-112 $$ |
| ⑤ | Combine like terms: $$ x^4+ \color{blue}{4x^3} \color{blue}{-11x^3} \color{red}{-44x^2} + \color{red}{32x^2} + \color{green}{128x} \color{green}{-28x} -112 = \\ = x^4 \color{blue}{-7x^3} \color{red}{-12x^2} + \color{green}{100x} -112 $$ |
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