$$\dfrac{1}{7500}(x-5)^4(x-2)(x+4)(x+6)=\dfrac{x^7-12x^6-6x^5+572x^4-1815x^3-4200x^2+26500x-30000}{7500}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{7500}(x-5)^4(x-2)(x+4)(x+6)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1}{7500}(x^4-20x^3+150x^2-500x+625)(x-2)(x+4)(x+6) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{x^4-20x^3+150x^2-500x+625}{7500}(x-2)(x+4)(x+6) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{x^5-22x^4+190x^3-800x^2+1625x-1250}{7500}(x+4)(x+6) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{x^6-18x^5+102x^4-40x^3-1575x^2+5250x-5000}{7500}(x+6) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{x^7-12x^6-6x^5+572x^4-1815x^3-4200x^2+26500x-30000}{7500}\end{aligned} $$ | |
| ① | $$ (x-5)^4 = (x-5)^2 \cdot (x-5)^2 $$ |
| ② | Find $ \left(x-5\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 5 }$. $$ \begin{aligned}\left(x-5\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 5 + \color{red}{5^2} = x^2-10x+25\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-10x+25}\right) $ by each term in $ \left( x^2-10x+25\right) $. $$ \left( \color{blue}{x^2-10x+25}\right) \cdot \left( x^2-10x+25\right) = x^4-10x^3+25x^2-10x^3+100x^2-250x+25x^2-250x+625 $$ |
| ④ | Combine like terms: $$ x^4 \color{blue}{-10x^3} + \color{red}{25x^2} \color{blue}{-10x^3} + \color{green}{100x^2} \color{orange}{-250x} + \color{green}{25x^2} \color{orange}{-250x} +625 = \\ = x^4 \color{blue}{-20x^3} + \color{green}{150x^2} \color{orange}{-500x} +625 $$ |
| ⑤ | Multiply $ \dfrac{1}{7500} $ by $ x^4-20x^3+150x^2-500x+625 $ to get $ \dfrac{ x^4-20x^3+150x^2-500x+625 }{ 7500 } $. Step 1: Write $ x^4-20x^3+150x^2-500x+625 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{7500} \cdot x^4-20x^3+150x^2-500x+625 & \xlongequal{\text{Step 1}} \frac{1}{7500} \cdot \frac{x^4-20x^3+150x^2-500x+625}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 1 \cdot \left( x^4-20x^3+150x^2-500x+625 \right) }{ 7500 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^4-20x^3+150x^2-500x+625 }{ 7500 } \end{aligned} $$ |
| ⑥ | Multiply $ \dfrac{x^4-20x^3+150x^2-500x+625}{7500} $ by $ x-2 $ to get $ \dfrac{x^5-22x^4+190x^3-800x^2+1625x-1250}{7500} $. 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. $$ \begin{aligned} \frac{x^4-20x^3+150x^2-500x+625}{7500} \cdot x-2 & \xlongequal{\text{Step 1}} \frac{x^4-20x^3+150x^2-500x+625}{7500} \cdot \frac{x-2}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^4-20x^3+150x^2-500x+625 \right) \cdot \left( x-2 \right) }{ 7500 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^5-2x^4-20x^4+40x^3+150x^3-300x^2-500x^2+1000x+625x-1250 }{ 7500 } = \\[1ex] &= \frac{x^5-22x^4+190x^3-800x^2+1625x-1250}{7500} \end{aligned} $$ |
| ⑦ | Multiply $ \dfrac{x^5-22x^4+190x^3-800x^2+1625x-1250}{7500} $ by $ x+4 $ to get $ \dfrac{x^6-18x^5+102x^4-40x^3-1575x^2+5250x-5000}{7500} $. Step 1: Write $ x+4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^5-22x^4+190x^3-800x^2+1625x-1250}{7500} \cdot x+4 & \xlongequal{\text{Step 1}} \frac{x^5-22x^4+190x^3-800x^2+1625x-1250}{7500} \cdot \frac{x+4}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^5-22x^4+190x^3-800x^2+1625x-1250 \right) \cdot \left( x+4 \right) }{ 7500 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^6+4x^5-22x^5-88x^4+190x^4+760x^3-800x^3-3200x^2+1625x^2+6500x-1250x-5000 }{ 7500 } = \frac{x^6-18x^5+102x^4-40x^3-1575x^2+5250x-5000}{7500} \end{aligned} $$ |
| ⑧ | Multiply $ \dfrac{x^6-18x^5+102x^4-40x^3-1575x^2+5250x-5000}{7500} $ by $ x+6 $ to get $ \dfrac{x^7-12x^6-6x^5+572x^4-1815x^3-4200x^2+26500x-30000}{7500} $. Step 1: Write $ x+6 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^6-18x^5+102x^4-40x^3-1575x^2+5250x-5000}{7500} \cdot x+6 & \xlongequal{\text{Step 1}} \frac{x^6-18x^5+102x^4-40x^3-1575x^2+5250x-5000}{7500} \cdot \frac{x+6}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^6-18x^5+102x^4-40x^3-1575x^2+5250x-5000 \right) \cdot \left( x+6 \right) }{ 7500 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^7+6x^6-18x^6-108x^5+102x^5+612x^4-40x^4-240x^3-1575x^3-9450x^2+5250x^2+31500x-5000x-30000 }{ 7500 } = \\[1ex] &= \frac{x^7-12x^6-6x^5+572x^4-1815x^3-4200x^2+26500x-30000}{7500} \end{aligned} $$ |