Question
Answer
$$\dfrac{4(x-2)(x-1)(x-3)(x-5)}{(4-2)\cdot(4-1)\cdot(4-3)\cdot(4-5)}=\dfrac{4x^4-44x^3+164x^2-244x+120}{-6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4(x-2)(x-1)(x-3)(x-5)}{(4-2)\cdot(4-1)\cdot(4-3)\cdot(4-5)}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{(4x-8)(x-1)(x-3)(x-5)}{\frac{64-48-16+12-32+24+8-6}{1}\cdot(-1)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{(4x^2-4x-8x+8)(x-3)(x-5)}{\frac{6}{1}\cdot(-1)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{(4x^2-12x+8)(x-3)(x-5)}{6\cdot(-1)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{(4x^3-12x^2-12x^2+36x+8x-24)(x-5)}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}\frac{(4x^3-24x^2+44x-24)(x-5)}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} } }}}\frac{4x^4-44x^3+164x^2-244x+120}{-6}\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{4} $ by $ \left( x-2\right) $ $$ \color{blue}{4} \cdot \left( x-2\right) = 4x-8 $$ |
| ② | $$ \color{blue}{ \left( 16-4-8 + 2\right) } \cdot \left( 4-3\right) = \color{blue}{16} \cdot4+\color{blue}{16} \cdot-3\color{blue}{-4} \cdot4\color{blue}{-4} \cdot-3\color{blue}{-8} \cdot4\color{blue}{-8} \cdot-3+\color{blue}{2} \cdot4+\color{blue}{2} \cdot-3 = \\ = 64-48-16 + 12-32 + 24 + 8-6 $$$$ \color{blue}{ 1 } \cdot 1 = 1 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{4x-8}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{4x-8}\right) \cdot \left( x-1\right) = 4x^2-4x-8x+8 $$ |
| ④ | Simplify numerator $$ \color{blue}{64} \color{red}{-48} \color{green}{-16} + \color{orange}{12} \color{blue}{-32} + \color{red}{24} + \color{green}{8} \color{green}{-6} = \color{green}{6} $$ |
| ⑤ | Combine like terms: $$ 4x^2 \color{blue}{-4x} \color{blue}{-8x} +8 = 4x^2 \color{blue}{-12x} +8 $$ |
| ⑥ | Remove 1 from denominator. |
| ⑦ | Multiply each term of $ \left( \color{blue}{4x^2-12x+8}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{4x^2-12x+8}\right) \cdot \left( x-3\right) = 4x^3-12x^2-12x^2+36x+8x-24 $$ |
| ⑧ | $ 6 \cdot ( -1 ) = -6 $ |
| ⑨ | Combine like terms: $$ 4x^3 \color{blue}{-12x^2} \color{blue}{-12x^2} + \color{red}{36x} + \color{red}{8x} -24 = 4x^3 \color{blue}{-24x^2} + \color{red}{44x} -24 $$ |
| ⑩ | $ 6 \cdot ( -1 ) = -6 $ |
| ⑪ | Multiply each term of $ \left( \color{blue}{4x^3-24x^2+44x-24}\right) $ by each term in $ \left( x-5\right) $. $$ \left( \color{blue}{4x^3-24x^2+44x-24}\right) \cdot \left( x-5\right) = 4x^4-20x^3-24x^3+120x^2+44x^2-220x-24x+120 $$ |
| ⑫ | Combine like terms: $$ 4x^4 \color{blue}{-20x^3} \color{blue}{-24x^3} + \color{red}{120x^2} + \color{red}{44x^2} \color{green}{-220x} \color{green}{-24x} +120 = \\ = 4x^4 \color{blue}{-44x^3} + \color{red}{164x^2} \color{green}{-244x} +120 $$ |
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