Question
Answer
$$x(x-1)(x-2)(x-3)+2x(x-1)(x-2)=x^4-4x^3+5x^2-2x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x(x-1)(x-2)(x-3)+2x(x-1)(x-2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-x)(x-2)(x-3)+(2x^2-2x)(x-2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-2x^2-x^2+2x)(x-3)+2x^3-4x^2-2x^2+4x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-3x^2+2x)(x-3)+2x^3-6x^2+4x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^4-3x^3-3x^3+9x^2+2x^2-6x+2x^3-6x^2+4x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-6x^3+11x^2-6x+2x^3-6x^2+4x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}x^4-4x^3+5x^2-2x\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{x} $ by $ \left( x-1\right) $ $$ \color{blue}{x} \cdot \left( x-1\right) = x^2-x $$Multiply $ \color{blue}{2x} $ by $ \left( x-1\right) $ $$ \color{blue}{2x} \cdot \left( x-1\right) = 2x^2-2x $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2-x}\right) $ by each term in $ \left( x-2\right) $. $$ \left( \color{blue}{x^2-x}\right) \cdot \left( x-2\right) = x^3-2x^2-x^2+2x $$Multiply each term of $ \left( \color{blue}{2x^2-2x}\right) $ by each term in $ \left( x-2\right) $. $$ \left( \color{blue}{2x^2-2x}\right) \cdot \left( x-2\right) = 2x^3-4x^2-2x^2+4x $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-2x^2} \color{blue}{-x^2} +2x = x^3 \color{blue}{-3x^2} +2x $$Combine like terms: $$ 2x^3 \color{blue}{-4x^2} \color{blue}{-2x^2} +4x = 2x^3 \color{blue}{-6x^2} +4x $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-3x^2+2x}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{x^3-3x^2+2x}\right) \cdot \left( x-3\right) = x^4-3x^3-3x^3+9x^2+2x^2-6x $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-3x^3} \color{blue}{-3x^3} + \color{red}{9x^2} + \color{red}{2x^2} -6x = x^4 \color{blue}{-6x^3} + \color{red}{11x^2} -6x $$ |
| ⑥ | Combine like terms: $$ x^4 \color{blue}{-6x^3} + \color{red}{11x^2} \color{green}{-6x} + \color{blue}{2x^3} \color{red}{-6x^2} + \color{green}{4x} = x^4 \color{blue}{-4x^3} + \color{red}{5x^2} \color{green}{-2x} $$ |
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