Answer
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{2}(x-2)(x+1)(x+2)(x+3)x& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{x-2}{2}(x+1)(x+2)(x+3)x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{x^2-x-2}{2}(x+2)(x+3)x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{x^3+x^2-4x-4}{2}(x+3)x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{x^4+4x^3-x^2-16x-12}{2}x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{x^5+4x^4-x^3-16x^2-12x}{2}\end{aligned} $$ | |
| ① | Multiply $ \dfrac{1}{2} $ by $ x-2 $ to get $ \dfrac{ x-2 }{ 2 } $. 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. $$ \begin{aligned} \frac{1}{2} \cdot x-2 & \xlongequal{\text{Step 1}} \frac{1}{2} \cdot \frac{x-2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot \left( x-2 \right) }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x-2 }{ 2 } \end{aligned} $$ |
| ② | Multiply $ \dfrac{x-2}{2} $ by $ x+1 $ to get $ \dfrac{x^2-x-2}{2} $. Step 1: Write $ x+1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x-2}{2} \cdot x+1 & \xlongequal{\text{Step 1}} \frac{x-2}{2} \cdot \frac{x+1}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x-2 \right) \cdot \left( x+1 \right) }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^2+x-2x-2 }{ 2 } = \frac{x^2-x-2}{2} \end{aligned} $$ |
| ③ | Multiply $ \dfrac{x^2-x-2}{2} $ by $ x+2 $ to get $ \dfrac{x^3+x^2-4x-4}{2} $. 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. $$ \begin{aligned} \frac{x^2-x-2}{2} \cdot x+2 & \xlongequal{\text{Step 1}} \frac{x^2-x-2}{2} \cdot \frac{x+2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x^2-x-2 \right) \cdot \left( x+2 \right) }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^3+2x^2-x^2-2x-2x-4 }{ 2 } = \frac{x^3+x^2-4x-4}{2} \end{aligned} $$ |
| ④ | Multiply $ \dfrac{x^3+x^2-4x-4}{2} $ by $ x+3 $ to get $ \dfrac{x^4+4x^3-x^2-16x-12}{2} $. 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. $$ \begin{aligned} \frac{x^3+x^2-4x-4}{2} \cdot x+3 & \xlongequal{\text{Step 1}} \frac{x^3+x^2-4x-4}{2} \cdot \frac{x+3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x^3+x^2-4x-4 \right) \cdot \left( x+3 \right) }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^4+3x^3+x^3+3x^2-4x^2-12x-4x-12 }{ 2 } = \frac{x^4+4x^3-x^2-16x-12}{2} \end{aligned} $$ |
| ⑤ | Multiply $ \dfrac{x^4+4x^3-x^2-16x-12}{2} $ by $ x $ to get $ \dfrac{ x^5+4x^4-x^3-16x^2-12x }{ 2 } $. Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^4+4x^3-x^2-16x-12}{2} \cdot x & \xlongequal{\text{Step 1}} \frac{x^4+4x^3-x^2-16x-12}{2} \cdot \frac{x}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^4+4x^3-x^2-16x-12 \right) \cdot x }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^5+4x^4-x^3-16x^2-12x }{ 2 } \end{aligned} $$ |