Question
Answer
$$\dfrac{5(x-2)(x-1)(x-3)(x-4)}{(5-2)\cdot(5-1)\cdot(5-3)\cdot(5-4)}=\dfrac{5x^4-50x^3+175x^2-250x+120}{24}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5(x-2)(x-1)(x-3)(x-4)}{(5-2)\cdot(5-1)\cdot(5-3)\cdot(5-4)}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{(5x-10)(x-1)(x-3)(x-4)}{\frac{125-75-25+15-50+30+10-6}{1}\cdot1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{(5x^2-5x-10x+10)(x-3)(x-4)}{\frac{24}{1}\cdot1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{(5x^2-15x+10)(x-3)(x-4)}{24\cdot1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{(5x^3-15x^2-15x^2+45x+10x-30)(x-4)}{24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}\frac{(5x^3-30x^2+55x-30)(x-4)}{24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}\frac{5x^4-50x^3+175x^2-250x+120}{24}\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{5} $ by $ \left( x-2\right) $ $$ \color{blue}{5} \cdot \left( x-2\right) = 5x-10 $$ |
| ② | $$ \color{blue}{ \left( 25-5-10 + 2\right) } \cdot \left( 5-3\right) = \color{blue}{25} \cdot5+\color{blue}{25} \cdot-3\color{blue}{-5} \cdot5\color{blue}{-5} \cdot-3\color{blue}{-10} \cdot5\color{blue}{-10} \cdot-3+\color{blue}{2} \cdot5+\color{blue}{2} \cdot-3 = \\ = 125-75-25 + 15-50 + 30 + 10-6 $$$$ \color{blue}{ 1 } \cdot 1 = 1 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{5x-10}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{5x-10}\right) \cdot \left( x-1\right) = 5x^2-5x-10x+10 $$ |
| ④ | Simplify numerator $$ \color{blue}{125} \color{red}{-75} \color{green}{-25} + \color{orange}{15} \color{blue}{-50} + \color{red}{30} + \color{green}{10} \color{green}{-6} = \color{green}{24} $$ |
| ⑤ | Combine like terms: $$ 5x^2 \color{blue}{-5x} \color{blue}{-10x} +10 = 5x^2 \color{blue}{-15x} +10 $$ |
| ⑥ | Remove 1 from denominator. |
| ⑦ | Multiply each term of $ \left( \color{blue}{5x^2-15x+10}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{5x^2-15x+10}\right) \cdot \left( x-3\right) = 5x^3-15x^2-15x^2+45x+10x-30 $$ |
| ⑧ | $ 24 \cdot 1 = 24 $ |
| ⑨ | Combine like terms: $$ 5x^3 \color{blue}{-15x^2} \color{blue}{-15x^2} + \color{red}{45x} + \color{red}{10x} -30 = 5x^3 \color{blue}{-30x^2} + \color{red}{55x} -30 $$ |
| ⑩ | $ 24 \cdot 1 = 24 $ |
| ⑪ | Multiply each term of $ \left( \color{blue}{5x^3-30x^2+55x-30}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{5x^3-30x^2+55x-30}\right) \cdot \left( x-4\right) = 5x^4-20x^3-30x^3+120x^2+55x^2-220x-30x+120 $$ |
| ⑫ | Combine like terms: $$ 5x^4 \color{blue}{-20x^3} \color{blue}{-30x^3} + \color{red}{120x^2} + \color{red}{55x^2} \color{green}{-220x} \color{green}{-30x} +120 = \\ = 5x^4 \color{blue}{-50x^3} + \color{red}{175x^2} \color{green}{-250x} +120 $$ |
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