Question
Answer
$$(x-3)^2(x-4)(x-1)=x^4-11x^3+43x^2-69x+36$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-3)^2(x-4)(x-1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-6x+9)(x-4)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-4x^2-6x^2+24x+9x-36)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-10x^2+33x-36)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-11x^3+43x^2-69x+36\end{aligned} $$ | |
| ① | Find $ \left(x-3\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}{ 3 }$. $$ \begin{aligned}\left(x-3\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 3 + \color{red}{3^2} = x^2-6x+9\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2-6x+9}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{x^2-6x+9}\right) \cdot \left( x-4\right) = x^3-4x^2-6x^2+24x+9x-36 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-4x^2} \color{blue}{-6x^2} + \color{red}{24x} + \color{red}{9x} -36 = x^3 \color{blue}{-10x^2} + \color{red}{33x} -36 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-10x^2+33x-36}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{x^3-10x^2+33x-36}\right) \cdot \left( x-1\right) = x^4-x^3-10x^3+10x^2+33x^2-33x-36x+36 $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-x^3} \color{blue}{-10x^3} + \color{red}{10x^2} + \color{red}{33x^2} \color{green}{-33x} \color{green}{-36x} +36 = x^4 \color{blue}{-11x^3} + \color{red}{43x^2} \color{green}{-69x} +36 $$ |
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