# How do you simplify (r^2+25) / (8r^3 -27) - ( 3r+2) /(2r-3)?

Dec 25, 2017

(-12r^3-25r^2-39r+7)/[(2r-3)(4r^2+6r+9)

#### Explanation:

First, factor $8 {r}^{3} - 27$ into $\left(2 r - 3\right) \left(4 {r}^{2} + 6 r + 9\right)$ using $\left({a}^{3} - {b}^{3}\right) = \left(a - b\right) \left({a}^{2} + a b + {b}^{2}\right)$.

We now get

$\frac{{r}^{2} + 25}{\left(2 r - 3\right) \left(4 {r}^{2} + 6 r + 9\right)} - \frac{3 r + 2}{2 r - 3}$

Multiply by $\left(4 {r}^{2} + 6 r + 9\right)$ in the second expression to get

[(3r+2)(4r^2+6r+9)]/[(2r-3)(4r^2+6r+9)

Now you can combine to get

{(r^2+25)-(3r+2)(4r^2+6r+9)}/[(2r-3)(4r^2+6r+9)

Expanding gives you

{(r^2+25)-(12r^3+26r^2+39r+18)}/[(2r-3)(4r^2+6r+9)

Last step is to simplify and you will get

(-12r^3-25r^2-39r+7)/[(2r-3)(4r^2+6r+9)