Pre-Calculus: Solving Rational Expressions

[adam denick]

f(x)= n(x)/d(x)
Always Factor f(x) and reduce first

Example equation:
y= (x^2 - 5x - 6)/(x+3)

So that will reduce to:
y= (x+1)(x-6)/(x+3)

Another equation:
y=(x^3 + 3x^2 - 4x - 12)/(x-5)

Reducing to:
y=(x+2)(x-2)(x+3)/(x-5)

Can someone give me a step by step to figure out the x-intercept(s), y-intercept, horizontal asymptotes, vertical asymptotes, and oblique aymptotes?

[bunny]

I am not a big fan of "go look it up" as an answer to a post. Nonetheless, I cant think you are ever going to get an answer here that is better than what you will find in an elementary college textbook on analytical calculus.

If you want an algorithm I would look there, I tried posting an answer here and it just got messy and confusing without purty pictures.

Half an answer would be-
1. Find y-int by putting x = 0
2. Find x-int by solving for y = 0
3. Find vertical asymptote by seeing where y is undefined
4. Find horizontal and oblique asymptotes by following procedure in a calculus textbook.

I do it in that order if that's any use

[adam denick]

I don't have my textbook with me, I'm not looking for complex answers, nor does it require a complex answer. For example, If I was asking how to solve to find out where the 'hole' in the graph is this is how I would need it answered.

So my example question is:
y=(x+4)(x-3)/(x+8)(x+4)

OK, so in order to find out where a hole in the graph is, there has to be an expression in both the numerator and denominator that cancel each other out. In this case, (x+4) cancels out, so the graph has a hole at -4.(although I'm not sure if this is the right answer for this)

This is the type of answer I'm looking for. I wish I could go and look it up, but oh well.

Edit: Thanks that helps a bit. Also if you could expand on the vertical asymptotes that would be appreciated. Anyone with any knowledge on horizontal and oblique aymptotes, thanks.

[chief444]

For the horizontal and oblique asymptotes it's simply reducing the equation down using polynomial division and finding the limits as x approaches infinity.

As a basic example (without actually doing your problem for you) say

y = x^2/(x-5)

With polynomial division we just add and subtract 25 to the top to get:

y = (x^2-25+25)/(x-5)

Reducing further we get:

y = [(x+5)(x-5)+25]/(x-5) = x+5+25/(x-5)

For large x the last term 25/(x-5) ~ 0 and becomes negligible and we're left with

y = x+5

And x+5 is an oblique asymptote.