Question
Answer
$$(x-1)^2(x-4)(x-1)=x^4-7x^3+15x^2-13x+4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-1)^2(x-4)(x-1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-2x+1)(x-4)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-4x^2-2x^2+8x+x-4)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-6x^2+9x-4)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-7x^3+15x^2-13x+4\end{aligned} $$ | |
| ① | Find $ \left(x-1\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}{ 1 }$. $$ \begin{aligned}\left(x-1\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 1 + \color{red}{1^2} = x^2-2x+1\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2-2x+1}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{x^2-2x+1}\right) \cdot \left( x-4\right) = x^3-4x^2-2x^2+8x+x-4 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-4x^2} \color{blue}{-2x^2} + \color{red}{8x} + \color{red}{x} -4 = x^3 \color{blue}{-6x^2} + \color{red}{9x} -4 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-6x^2+9x-4}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{x^3-6x^2+9x-4}\right) \cdot \left( x-1\right) = x^4-x^3-6x^3+6x^2+9x^2-9x-4x+4 $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-x^3} \color{blue}{-6x^3} + \color{red}{6x^2} + \color{red}{9x^2} \color{green}{-9x} \color{green}{-4x} +4 = x^4 \color{blue}{-7x^3} + \color{red}{15x^2} \color{green}{-13x} +4 $$ |
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