Question
Answer
$$(x-2)^2(x-4)(x-6)=x^4-14x^3+68x^2-136x+96$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-2)^2(x-4)(x-6)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-4x+4)(x-4)(x-6) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-4x^2-4x^2+16x+4x-16)(x-6) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-8x^2+20x-16)(x-6) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-14x^3+68x^2-136x+96\end{aligned} $$ | |
| 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^2-4x+4}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{x^2-4x+4}\right) \cdot \left( x-4\right) = x^3-4x^2-4x^2+16x+4x-16 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-4x^2} \color{blue}{-4x^2} + \color{red}{16x} + \color{red}{4x} -16 = x^3 \color{blue}{-8x^2} + \color{red}{20x} -16 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-8x^2+20x-16}\right) $ by each term in $ \left( x-6\right) $. $$ \left( \color{blue}{x^3-8x^2+20x-16}\right) \cdot \left( x-6\right) = x^4-6x^3-8x^3+48x^2+20x^2-120x-16x+96 $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-6x^3} \color{blue}{-8x^3} + \color{red}{48x^2} + \color{red}{20x^2} \color{green}{-120x} \color{green}{-16x} +96 = \\ = x^4 \color{blue}{-14x^3} + \color{red}{68x^2} \color{green}{-136x} +96 $$ |
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