$$-\dfrac{125}{864}(x-6)^2(x-2)(x-3)(x-8)(x^2+1)=\dfrac{125x^7-3125x^6+29875x^5-136625x^4+308750x^3-349500x^2+279000x-216000}{864}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-\frac{125}{864}(x-6)^2(x-2)(x-3)(x-8)(x^2+1)& \xlongequal{ }-\frac{125}{864}(x^2-12x+36)(x-2)(x-3)(x-8)(x^2+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{125x^2-1500x+4500}{864}(x-2)(x-3)(x-8)(x^2+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{125x^3-1750x^2+7500x-9000}{864}(x-3)(x-8)(x^2+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{125x^4-2125x^3+12750x^2-31500x+27000}{864}(x-8)(x^2+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{125x^5-3125x^4+29750x^3-133500x^2+279000x-216000}{864}(x^2+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{125x^7-3125x^6+29875x^5-136625x^4+308750x^3-349500x^2+279000x-216000}{864}\end{aligned} $$ | |
| ① | Multiply $ \dfrac{125}{864} $ by $ x^2-12x+36 $ to get $ \dfrac{ 125x^2-1500x+4500 }{ 864 } $. Step 1: Write $ x^2-12x+36 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{125}{864} \cdot x^2-12x+36 & \xlongequal{\text{Step 1}} \frac{125}{864} \cdot \frac{x^2-12x+36}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 125 \cdot \left( x^2-12x+36 \right) }{ 864 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 125x^2-1500x+4500 }{ 864 } \end{aligned} $$ |
| ② | Multiply $ \dfrac{125x^2-1500x+4500}{864} $ by $ x-2 $ to get $ \dfrac{125x^3-1750x^2+7500x-9000}{864} $. Step 1: Write $ x-2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{125x^2-1500x+4500}{864} \cdot x-2 & \xlongequal{\text{Step 1}} \frac{125x^2-1500x+4500}{864} \cdot \frac{x-2}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 125x^2-1500x+4500 \right) \cdot \left( x-2 \right) }{ 864 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 125x^3-250x^2-1500x^2+3000x+4500x-9000 }{ 864 } = \\[1ex] &= \frac{125x^3-1750x^2+7500x-9000}{864} \end{aligned} $$ |
| ③ | Multiply $ \dfrac{125x^3-1750x^2+7500x-9000}{864} $ by $ x-3 $ to get $ \dfrac{125x^4-2125x^3+12750x^2-31500x+27000}{864} $. Step 1: Write $ x-3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{125x^3-1750x^2+7500x-9000}{864} \cdot x-3 & \xlongequal{\text{Step 1}} \frac{125x^3-1750x^2+7500x-9000}{864} \cdot \frac{x-3}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 125x^3-1750x^2+7500x-9000 \right) \cdot \left( x-3 \right) }{ 864 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 125x^4-375x^3-1750x^3+5250x^2+7500x^2-22500x-9000x+27000 }{ 864 } = \\[1ex] &= \frac{125x^4-2125x^3+12750x^2-31500x+27000}{864} \end{aligned} $$ |
| ④ | Multiply $ \dfrac{125x^4-2125x^3+12750x^2-31500x+27000}{864} $ by $ x-8 $ to get $ \dfrac{125x^5-3125x^4+29750x^3-133500x^2+279000x-216000}{864} $. Step 1: Write $ x-8 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{125x^4-2125x^3+12750x^2-31500x+27000}{864} \cdot x-8 & \xlongequal{\text{Step 1}} \frac{125x^4-2125x^3+12750x^2-31500x+27000}{864} \cdot \frac{x-8}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 125x^4-2125x^3+12750x^2-31500x+27000 \right) \cdot \left( x-8 \right) }{ 864 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 125x^5-1000x^4-2125x^4+17000x^3+12750x^3-102000x^2-31500x^2+252000x+27000x-216000 }{ 864 } = \frac{125x^5-3125x^4+29750x^3-133500x^2+279000x-216000}{864} \end{aligned} $$ |
| ⑤ | Multiply $ \dfrac{125x^5-3125x^4+29750x^3-133500x^2+279000x-216000}{864} $ by $ x^2+1 $ to get $ \dfrac{125x^7-3125x^6+29875x^5-136625x^4+308750x^3-349500x^2+279000x-216000}{864} $. Step 1: Write $ x^2+1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{125x^5-3125x^4+29750x^3-133500x^2+279000x-216000}{864} \cdot x^2+1 & \xlongequal{\text{Step 1}} \frac{125x^5-3125x^4+29750x^3-133500x^2+279000x-216000}{864} \cdot \frac{x^2+1}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 125x^5-3125x^4+29750x^3-133500x^2+279000x-216000 \right) \cdot \left( x^2+1 \right) }{ 864 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 125x^7+125x^5-3125x^6-3125x^4+29750x^5+29750x^3-133500x^4-133500x^2+279000x^3+279000x-216000x^2-216000 }{ 864 } = \\[1ex] &= \frac{125x^7-3125x^6+29875x^5-136625x^4+308750x^3-349500x^2+279000x-216000}{864} \end{aligned} $$ |