$$\dfrac{3(x-2)(x-1)(x-4)(x-5)}{(3-2)\cdot(3-1)\cdot(3-4)\cdot(3-5)}=\dfrac{3x^4-36x^3+147x^2-234x+120}{4}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3(x-2)(x-1)(x-4)(x-5)}{(3-2)\cdot(3-1)\cdot(3-4)\cdot(3-5)}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{(3x-6)(x-1)(x-4)(x-5)}{\frac{27-36-9+12-18+24+6-8}{1}\cdot(-2)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{(3x^2-3x-6x+6)(x-4)(x-5)}{(-\frac{2}{1})\cdot(-2)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{(3x^2-9x+6)(x-4)(x-5)}{4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{(3x^3-12x^2-9x^2+36x+6x-24)(x-5)}{4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}\frac{(3x^3-21x^2+42x-24)(x-5)}{4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}\frac{3x^4-36x^3+147x^2-234x+120}{4}\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{3} $ by $ \left( x-2\right) $ $$ \color{blue}{3} \cdot \left( x-2\right) = 3x-6 $$ |
| ② | $$ \color{blue}{ \left( 9-3-6 + 2\right) } \cdot \left( 3-4\right) = \color{blue}{9} \cdot3+\color{blue}{9} \cdot-4\color{blue}{-3} \cdot3\color{blue}{-3} \cdot-4\color{blue}{-6} \cdot3\color{blue}{-6} \cdot-4+\color{blue}{2} \cdot3+\color{blue}{2} \cdot-4 = \\ = 27-36-9 + 12-18 + 24 + 6-8 $$$$ \color{blue}{ 1 } \cdot 1 = 1 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{3x-6}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{3x-6}\right) \cdot \left( x-1\right) = 3x^2-3x-6x+6 $$ |
| ④ | Simplify numerator $$ \color{blue}{27} \color{red}{-36} \color{green}{-9} + \color{orange}{12} \color{blue}{-18} + \color{red}{24} + \color{green}{6} \color{green}{-8} = \color{green}{-2} $$ |
| ⑤ | Combine like terms: $$ 3x^2 \color{blue}{-3x} \color{blue}{-6x} +6 = 3x^2 \color{blue}{-9x} +6 $$ |
| ⑥ | Multiply $ \dfrac{-2}{1} $ by $ -2 $ to get $ 4$. Write $ -2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. $$ \begin{aligned} \frac{-2}{1} \cdot -2 = \frac{-2}{1} \cdot \frac{-2}{\color{red}{1}} = \frac{4}{1} =4 \end{aligned} $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{3x^2-9x+6}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{3x^2-9x+6}\right) \cdot \left( x-4\right) = 3x^3-12x^2-9x^2+36x+6x-24 $$ |
| ⑧ | Multiply $ \dfrac{-2}{1} $ by $ -2 $ to get $ 4$. Write $ -2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. $$ \begin{aligned} \frac{-2}{1} \cdot -2 = \frac{-2}{1} \cdot \frac{-2}{\color{red}{1}} = \frac{4}{1} =4 \end{aligned} $$ |
| ⑨ | Combine like terms: $$ 3x^3 \color{blue}{-12x^2} \color{blue}{-9x^2} + \color{red}{36x} + \color{red}{6x} -24 = 3x^3 \color{blue}{-21x^2} + \color{red}{42x} -24 $$ |
| ⑩ | Multiply $ \dfrac{-2}{1} $ by $ -2 $ to get $ 4$. Write $ -2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. $$ \begin{aligned} \frac{-2}{1} \cdot -2 = \frac{-2}{1} \cdot \frac{-2}{\color{red}{1}} = \frac{4}{1} =4 \end{aligned} $$ |
| ⑪ | Multiply each term of $ \left( \color{blue}{3x^3-21x^2+42x-24}\right) $ by each term in $ \left( x-5\right) $. $$ \left( \color{blue}{3x^3-21x^2+42x-24}\right) \cdot \left( x-5\right) = 3x^4-15x^3-21x^3+105x^2+42x^2-210x-24x+120 $$ |
| ⑫ | Combine like terms: $$ 3x^4 \color{blue}{-15x^3} \color{blue}{-21x^3} + \color{red}{105x^2} + \color{red}{42x^2} \color{green}{-210x} \color{green}{-24x} +120 = \\ = 3x^4 \color{blue}{-36x^3} + \color{red}{147x^2} \color{green}{-234x} +120 $$ |