# How do you combine (3z)/(z-3)-z/(z+4)?

Aug 22, 2016

$\frac{z \left(2 z + 15\right)}{\left(z - 3\right) \left(z + 4\right)}$.

#### Explanation:

The Expression$= \frac{3 z}{z - 3} - \frac{z}{z + 4}$

$= \frac{3 z \left(z + 4\right) - z \left(z - 3\right)}{\left(z - 3\right) \left(z + 4\right)}$

$= \frac{3 {z}^{2} + 12 z - {z}^{2} + 3 z}{\left(z - 3\right) \left(z + 4\right)}$

$= \frac{2 {z}^{2} + 15 z}{\left(z - 3\right) \left(z + 4\right)}$

$= \frac{z \left(2 z + 15\right)}{\left(z - 3\right) \left(z + 4\right)}$.

Aug 22, 2016

$\frac{3 z \left(z + 4\right) - z \left(z - 3\right)}{\left(z + 4\right) \left(z - 3\right)}$

Or you can expand the brackets. I will let you do that

#### Explanation:

color(blue)("Consider: "(3z)/(z-3)

Multiply by 1 but in the form of $1 = \frac{z + 4}{z + 4}$ giving

(3z)/(z-3)xx(z+4)/(z+4) = (3z(z+4))/((z-3)(z+4)

'...........................................................................

color(blue)("Consider: "-z/(z+4)

Multiply by 1 but in the form $1 = \frac{z - 3}{z - 3}$

$- \frac{z}{z + 4} \times \frac{z - 3}{z - 3} = - \frac{z \left(z - 3\right)}{\left(z + 4\right) \left(z - 3\right)}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Putting it all together}}$

$\frac{3 z \left(z + 4\right)}{\left(z - 3\right) \left(z + 4\right)} - \frac{z \left(z - 3\right)}{\left(z + 4\right) \left(z - 3\right)}$

$\frac{3 z \left(z + 4\right) - z \left(z - 3\right)}{\left(z + 4\right) \left(z - 3\right)}$

Or you can expand the brackets. I will let you do that.