Simplifying an Algebraic Fraction

Date: 05/09/2000 at 12:49:43
From: M. Hancock
Subject: Algebraic complex fractions

Dear Dr. Math:

I am retaking algebra with my 7th grade son (we are homeschoolers), 
and I am stuck on this type of problem:

     1 + 1/(x + 1/x)

I am to simplify this. The answer given in the book is:

     (x^2 + x + 1)/(x^2 + 1)

I have tried x as a common denominator with the one just hanging out 
there. I get the denominator, but can't seem to get the x term in the 
numerator. The book calls this "complex fractions."

M. Hancock

Date: 05/09/2000 at 16:24:01
From: Doctor Wilkinson
Subject: Re: Algebraic complex fractions

Hi.

Let's take it step by step. Ignoring the first term for the time 
being, we have:

         1
     ---------
      x + 1/x

Let's start by simplifying the denominator. That means putting it over 
a common denominator x (you were doing fine so far). This should give 
you a denominator of:

      x^2 + 1
     ---------
         x

Now you have 1 over that, so flip it upside down to get:

         x
     ---------
      x^2 + 1

Now you have 1 + that, so you want now to make x^2 + 1 the common 
denominator, which gives you:

      x^2 + 1 + x
     -------------
        x^2 + 1

or:

      x^2 + x + 1
     -------------
        x^2 + 1

- Doctor Wilkinson, The Math Forum
  http://mathforum.org/dr.math/