$$\dfrac{1(x-2)(x-3)(x-4)(x-5)}{(1-2)\cdot(1-3)\cdot(1-4)\cdot(1-5)}=\dfrac{x^4-14x^3+71x^2-154x+120}{24}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1(x-2)(x-3)(x-4)(x-5)}{(1-2)\cdot(1-3)\cdot(1-4)\cdot(1-5)}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{(x-2)(x-3)(x-4)(x-5)}{\frac{1-4-3+12-2+8+6-24}{1}\cdot(-4)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{(x^2-3x-2x+6)(x-4)(x-5)}{(-\frac{6}{1})\cdot(-4)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{(x^2-5x+6)(x-4)(x-5)}{24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{(x^3-4x^2-5x^2+20x+6x-24)(x-5)}{24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}\frac{(x^3-9x^2+26x-24)(x-5)}{24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}\frac{x^4-14x^3+71x^2-154x+120}{24}\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{1} $ by $ \left( x-2\right) $ $$ \color{blue}{1} \cdot \left( x-2\right) = x-2 $$ |
| ② | $$ \color{blue}{ \left( 1-3-2 + 6\right) } \cdot \left( 1-4\right) = \color{blue}{1} \cdot1+\color{blue}{1} \cdot-4\color{blue}{-3} \cdot1\color{blue}{-3} \cdot-4\color{blue}{-2} \cdot1\color{blue}{-2} \cdot-4+\color{blue}{6} \cdot1+\color{blue}{6} \cdot-4 = \\ = 1-4-3 + 12-2 + 8 + 6-24 $$$$ \color{blue}{ 1 } \cdot 1 = 1 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x-2}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{x-2}\right) \cdot \left( x-3\right) = x^2-3x-2x+6 $$ |
| ④ | Simplify numerator $$ \color{blue}{1} \color{red}{-4} \color{green}{-3} + \color{orange}{12} \color{blue}{-2} + \color{red}{8} + \color{green}{6} \color{green}{-24} = \color{green}{-6} $$ |
| ⑤ | Combine like terms: $$ x^2 \color{blue}{-3x} \color{blue}{-2x} +6 = x^2 \color{blue}{-5x} +6 $$ |
| ⑥ | Multiply $ \dfrac{-6}{1} $ by $ -4 $ to get $ 24$. Write $ -4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. $$ \begin{aligned} \frac{-6}{1} \cdot -4 = \frac{-6}{1} \cdot \frac{-4}{\color{red}{1}} = \frac{24}{1} =24 \end{aligned} $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{x^2-5x+6}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{x^2-5x+6}\right) \cdot \left( x-4\right) = x^3-4x^2-5x^2+20x+6x-24 $$ |
| ⑧ | Multiply $ \dfrac{-6}{1} $ by $ -4 $ to get $ 24$. Write $ -4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. $$ \begin{aligned} \frac{-6}{1} \cdot -4 = \frac{-6}{1} \cdot \frac{-4}{\color{red}{1}} = \frac{24}{1} =24 \end{aligned} $$ |
| ⑨ | Combine like terms: $$ x^3 \color{blue}{-4x^2} \color{blue}{-5x^2} + \color{red}{20x} + \color{red}{6x} -24 = x^3 \color{blue}{-9x^2} + \color{red}{26x} -24 $$ |
| ⑩ | Multiply $ \dfrac{-6}{1} $ by $ -4 $ to get $ 24$. Write $ -4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. $$ \begin{aligned} \frac{-6}{1} \cdot -4 = \frac{-6}{1} \cdot \frac{-4}{\color{red}{1}} = \frac{24}{1} =24 \end{aligned} $$ |
| ⑪ | Multiply each term of $ \left( \color{blue}{x^3-9x^2+26x-24}\right) $ by each term in $ \left( x-5\right) $. $$ \left( \color{blue}{x^3-9x^2+26x-24}\right) \cdot \left( x-5\right) = x^4-5x^3-9x^3+45x^2+26x^2-130x-24x+120 $$ |
| ⑫ | Combine like terms: $$ x^4 \color{blue}{-5x^3} \color{blue}{-9x^3} + \color{red}{45x^2} + \color{red}{26x^2} \color{green}{-130x} \color{green}{-24x} +120 = \\ = x^4 \color{blue}{-14x^3} + \color{red}{71x^2} \color{green}{-154x} +120 $$ |