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Question

Answer

$$5\cdot\dfrac{d}{d^2+48d+5}+d+2=\dfrac{d^3+50d^2+106d+10}{d^2+48d+5}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}5 \cdot \frac{d}{d^2+48d+5}+d+2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5d}{d^2+48d+5}+d+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{d^3+48d^2+10d}{d^2+48d+5}+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{d^3+50d^2+106d+10}{d^2+48d+5}\end{aligned} $$
①

Multiply $5$ by $ \dfrac{d}{d^2+48d+5} $ to get $ \dfrac{ 5d }{ d^2+48d+5 } $.

Step 1: Write $ 5 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 5 \cdot \frac{d}{d^2+48d+5} & \xlongequal{\text{Step 1}} \frac{5}{\color{red}{1}} \cdot \frac{d}{d^2+48d+5} \xlongequal{\text{Step 2}} \frac{ 5 \cdot d }{ 1 \cdot \left( d^2+48d+5 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 5d }{ d^2+48d+5 } \end{aligned} $$
②

Add $ \dfrac{5d}{d^2+48d+5} $ and $ d $ to get $ \dfrac{ \color{purple}{ d^3+48d^2+10d } }{ d^2+48d+5 }$.

Step 1: Write $ d $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{d^2+48d+5}$.

$$ \begin{aligned} \frac{5d}{d^2+48d+5} +d & \xlongequal{\text{Step 1}} \frac{5d}{d^2+48d+5} + \frac{d}{\color{red}{1}} = \frac{ 5d }{ d^2+48d+5 } + \frac{ d \cdot \color{blue}{ \left( d^2+48d+5 \right) }}{ 1 \cdot \color{blue}{ \left( d^2+48d+5 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 5d } }{ d^2+48d+5 } + \frac{ \color{purple}{ d^3+48d^2+5d } }{ d^2+48d+5 }=\frac{ \color{purple}{ d^3+48d^2+10d } }{ d^2+48d+5 } \end{aligned} $$
③

Add $ \dfrac{d^3+48d^2+10d}{d^2+48d+5} $ and $ 2 $ to get $ \dfrac{ \color{purple}{ d^3+50d^2+106d+10 } }{ d^2+48d+5 }$.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{d^2+48d+5}$.

$$ \begin{aligned} \frac{d^3+48d^2+10d}{d^2+48d+5} +2 & \xlongequal{\text{Step 1}} \frac{d^3+48d^2+10d}{d^2+48d+5} + \frac{2}{\color{red}{1}} = \frac{ d^3+48d^2+10d }{ d^2+48d+5 } + \frac{ 2 \cdot \color{blue}{ \left( d^2+48d+5 \right) }}{ 1 \cdot \color{blue}{ \left( d^2+48d+5 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ d^3+48d^2+10d } }{ d^2+48d+5 } + \frac{ \color{purple}{ 2d^2+96d+10 } }{ d^2+48d+5 } = \\[1ex] &=\frac{ \color{purple}{ d^3+50d^2+106d+10 } }{ d^2+48d+5 } \end{aligned} $$
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