Question
Answer
$$(x-6)(x-2)^2(x+4)=x^4-6x^3-12x^2+88x-96$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-6)(x-2)^2(x+4)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x-6)(x^2-4x+4)(x+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-4x^2+4x-6x^2+24x-24)(x+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-10x^2+28x-24)(x+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-6x^3-12x^2+88x-96\end{aligned} $$ | |
| 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-6}\right) $ by each term in $ \left( x^2-4x+4\right) $. $$ \left( \color{blue}{x-6}\right) \cdot \left( x^2-4x+4\right) = x^3-4x^2+4x-6x^2+24x-24 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-4x^2} + \color{red}{4x} \color{blue}{-6x^2} + \color{red}{24x} -24 = x^3 \color{blue}{-10x^2} + \color{red}{28x} -24 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-10x^2+28x-24}\right) $ by each term in $ \left( x+4\right) $. $$ \left( \color{blue}{x^3-10x^2+28x-24}\right) \cdot \left( x+4\right) = x^4+4x^3-10x^3-40x^2+28x^2+112x-24x-96 $$ |
| ⑤ | Combine like terms: $$ x^4+ \color{blue}{4x^3} \color{blue}{-10x^3} \color{red}{-40x^2} + \color{red}{28x^2} + \color{green}{112x} \color{green}{-24x} -96 = x^4 \color{blue}{-6x^3} \color{red}{-12x^2} + \color{green}{88x} -96 $$ |
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