Solve z^2-64/z+z+8/z+4

$$\frac { z ^ { 2 } - 64 } { z } + \frac { z + 8 } { z + 4 }$$

Evaluate

$$\frac{\left(z+8\right)\left(z^{2}-3z-32\right)}{z\left(z+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of z and z+4 is z\left(z+4\right). Multiply \frac{z^{2}-64}{z} times \frac{z+4}{z+4}. Multiply \frac{z+8}{z+4} times \frac{z}{z}.

$$\frac{\left(z^{2}-64\right)\left(z+4\right)}{z\left(z+4\right)}+\frac{\left(z+8\right)z}{z\left(z+4\right)}$$

Since \frac{\left(z^{2}-64\right)\left(z+4\right)}{z\left(z+4\right)} and \frac{\left(z+8\right)z}{z\left(z+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(z^{2}-64\right)\left(z+4\right)+\left(z+8\right)z}{z\left(z+4\right)}$$

Do the multiplications in \left(z^{2}-64\right)\left(z+4\right)+\left(z+8\right)z.

$$\frac{z^{3}+4z^{2}-64z-256+z^{2}+8z}{z\left(z+4\right)}$$

Combine like terms in z^{3}+4z^{2}-64z-256+z^{2}+8z.

$$\frac{z^{3}+5z^{2}-56z-256}{z\left(z+4\right)}$$

Expand z\left(z+4\right).

$$\frac{z^{3}+5z^{2}-56z-256}{z^{2}+4z}$$

Expand

$$\frac{z^{3}+5z^{2}-56z-256}{z\left(z+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of z and z+4 is z\left(z+4\right). Multiply \frac{z^{2}-64}{z} times \frac{z+4}{z+4}. Multiply \frac{z+8}{z+4} times \frac{z}{z}.

$$\frac{\left(z^{2}-64\right)\left(z+4\right)}{z\left(z+4\right)}+\frac{\left(z+8\right)z}{z\left(z+4\right)}$$

Since \frac{\left(z^{2}-64\right)\left(z+4\right)}{z\left(z+4\right)} and \frac{\left(z+8\right)z}{z\left(z+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(z^{2}-64\right)\left(z+4\right)+\left(z+8\right)z}{z\left(z+4\right)}$$

Do the multiplications in \left(z^{2}-64\right)\left(z+4\right)+\left(z+8\right)z.

$$\frac{z^{3}+4z^{2}-64z-256+z^{2}+8z}{z\left(z+4\right)}$$

Combine like terms in z^{3}+4z^{2}-64z-256+z^{2}+8z.

$$\frac{z^{3}+5z^{2}-56z-256}{z\left(z+4\right)}$$

Expand z\left(z+4\right).

$$\frac{z^{3}+5z^{2}-56z-256}{z^{2}+4z}$$