Add $z^2$ and $ \dfrac{19}{z^2} $ to get $ \dfrac{ \color{purple}{ z^4+19 } }{ z^2 }$.

Step 1: Write $ z^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 first fraction by $\color{blue}{ z^2 }$.

$$ \begin{aligned} z^2+ \frac{19}{z^2} & \xlongequal{\text{Step 1}} \frac{z^2}{\color{red}{1}} + \frac{19}{z^2} = \frac{ z^2 \cdot \color{blue}{ z^2 }}{ 1 \cdot \color{blue}{ z^2 }} + \frac{ 19 }{ z^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ z^4 } }{ z^2 } + \frac{ \color{purple}{ 19 } }{ z^2 }=\frac{ \color{purple}{ z^4+19 } }{ z^2 } \end{aligned} $$

Subtract $z$ from $ \dfrac{z^4+19}{z^2} $ to get $ \dfrac{ \color{purple}{ z^4-z^3+19 } }{ z^2 }$.

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

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

$$ \begin{aligned} \frac{z^4+19}{z^2} -z & \xlongequal{\text{Step 1}} \frac{z^4+19}{z^2} - \frac{z}{\color{red}{1}} = \frac{ z^4+19 }{ z^2 } - \frac{ z \cdot \color{blue}{ z^2 }}{ 1 \cdot \color{blue}{ z^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ z^4+19 } }{ z^2 } - \frac{ \color{purple}{ z^3 } }{ z^2 }=\frac{ \color{purple}{ z^4-z^3+19 } }{ z^2 } \end{aligned} $$

Subtract $2$ from $ \dfrac{z^4-z^3+19}{z^2} $ to get $ \dfrac{ \color{purple}{ z^4-z^3-2z^2+19 } }{ z^2 }$.

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

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

$$ \begin{aligned} \frac{z^4-z^3+19}{z^2} -2 & \xlongequal{\text{Step 1}} \frac{z^4-z^3+19}{z^2} - \frac{2}{\color{red}{1}} = \frac{ z^4-z^3+19 }{ z^2 } - \frac{ 2 \cdot \color{blue}{ z^2 }}{ 1 \cdot \color{blue}{ z^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ z^4-z^3+19 } }{ z^2 } - \frac{ \color{purple}{ 2z^2 } }{ z^2 }=\frac{ \color{purple}{ z^4-z^3-2z^2+19 } }{ z^2 } \end{aligned} $$