Tap the blue circles to see an explanation.
| $$ \begin{aligned}-\frac{1}{1000}(x-80)(x-90)(x-85)(x-60)+30& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{x-80}{1000}(x-90)(x-85)(x-60)+30 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{x^2-170x+7200}{1000}(x-85)(x-60)+30 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{x^3-255x^2+21650x-612000}{1000}(x-60)+30 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{x^4-315x^3+36950x^2-1911000x+36720000}{1000}+30 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{x^4-315x^3+36950x^2-1911000x+36750000}{1000}\end{aligned} $$ | |
| ① | Multiply $ \dfrac{1}{1000} $ by $ x-80 $ to get $ \dfrac{ x-80 }{ 1000 } $. Step 1: Write $ x-80 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{1000} \cdot x-80 & \xlongequal{\text{Step 1}} \frac{1}{1000} \cdot \frac{x-80}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot \left( x-80 \right) }{ 1000 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x-80 }{ 1000 } \end{aligned} $$ |
| ② | Multiply $ \dfrac{x-80}{1000} $ by $ x-90 $ to get $ \dfrac{x^2-170x+7200}{1000} $. Step 1: Write $ x-90 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x-80}{1000} \cdot x-90 & \xlongequal{\text{Step 1}} \frac{x-80}{1000} \cdot \frac{x-90}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x-80 \right) \cdot \left( x-90 \right) }{ 1000 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^2-90x-80x+7200 }{ 1000 } = \frac{x^2-170x+7200}{1000} \end{aligned} $$ |
| ③ | Multiply $ \dfrac{x^2-170x+7200}{1000} $ by $ x-85 $ to get $ \dfrac{x^3-255x^2+21650x-612000}{1000} $. Step 1: Write $ x-85 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^2-170x+7200}{1000} \cdot x-85 & \xlongequal{\text{Step 1}} \frac{x^2-170x+7200}{1000} \cdot \frac{x-85}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^2-170x+7200 \right) \cdot \left( x-85 \right) }{ 1000 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^3-85x^2-170x^2+14450x+7200x-612000 }{ 1000 } = \\[1ex] &= \frac{x^3-255x^2+21650x-612000}{1000} \end{aligned} $$ |
| ④ | Multiply $ \dfrac{x^3-255x^2+21650x-612000}{1000} $ by $ x-60 $ to get $ \dfrac{x^4-315x^3+36950x^2-1911000x+36720000}{1000} $. Step 1: Write $ x-60 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^3-255x^2+21650x-612000}{1000} \cdot x-60 & \xlongequal{\text{Step 1}} \frac{x^3-255x^2+21650x-612000}{1000} \cdot \frac{x-60}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^3-255x^2+21650x-612000 \right) \cdot \left( x-60 \right) }{ 1000 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^4-60x^3-255x^3+15300x^2+21650x^2-1299000x-612000x+36720000 }{ 1000 } = \\[1ex] &= \frac{x^4-315x^3+36950x^2-1911000x+36720000}{1000} \end{aligned} $$ |
| ⑤ | Add $ \dfrac{x^4-315x^3+36950x^2-1911000x+36720000}{1000} $ and $ 30 $ to get $ \dfrac{ \color{purple}{ x^4-315x^3+36950x^2-1911000x+36750000 } }{ 1000 }$. Step 1: Write $ 30 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To add raitonal expressions, both fractions must have the same denominator. |