Rational Expression With Two Variables

I'm a college student teaching himself algebra (loooong story) to save money on classes. I've done well so far, just finished up polynomials and now moving on to rational expressions. I've got the basics of factoring and writing in simplest terms down, but my book threw me a curveball:

Write in simplest terms:

9r^{2} - 4s^{2}

-------

9r + 6s

I've factored it out to:

3r(3 + r) - 2s(2 + s)

----------

3(3 + r) + 3(2 + s)

Crossing out like terms, this leaves me with

3r-2s

-------

6

However, the book says the denominator should be 3, not 6. Knowing the answer doesn't do me any good if I don't understand how to get there. Where am I going wrong in solving this?

Thanks!

Re: Rational Expression With Two Variables

We are given to simplify:

$\displaystyle \frac{9r^2-4s^2}{9r+6s}$

Now, factor first, using a difference of squares for the numerator:

$\displaystyle \frac{(3r+2s)(3r-2s)}{3(3r+2s)}$

Cancel or divide out common factors to get:

$\displaystyle \frac{3r-2s}{3}$

Re: Rational Expression With Two Variables

Wow, I just got done covering factoring too. Thanks so much!