Expert: Steve Holleran - 3/13/2007

Question
preform indicated operations and simplify:

x/x+2 + 1/x-3 - x^2-2/x^2-x-6

Answer
Hello Mallory,

Okay, since you're combining fractions here, you want to find the common denominator, or LCD.  First,though, all of the denominators have to be in factored form.  The first two parts of the expression are okay, but the third denominator, x^2 - 2x -6, has to be factored into :

x^2 -2x -6 = (x - 3)(x + 2). Now the expression looks like:

  x / (x+2) + 1 / (x-3) - (x^-2) / (x-3)(x+2)

So, in the LCD, you need a factor of (x+2) and a factor of (x-3).  Just write their product, don't multiply them out:

LCD = (x+2)(x-3)

Now to get each fraction over the LCD, you have to multiply the numerator of each by whatever factor in the LCD is missing from the denominator.

In the first fraction, the denominator is (x+2).  If you look at the LCD, the (x-3) factor is missing, so you have to multiply top and bottom by this:

 x / (x+2) * (x-3) / (x-3) = x (x -3) / (x+2)(x-3)

For the second fraction, the denominator is (x-3).  If you look at the LCD, the (x+2) is missing, so you have to multiply top and bottom by this:

 1 / (x-3) * (x+2) / (x+2) = 1(x+2) / (x-3)(x+2)

For the third fraction, the denominator is exactly the same as the LCD, so just multiply it by 1:

 - (x^2-2) / (x-3)(x+2) = -1(x^2-2) / (x-3)(x+2).

Now, since they all have the same denominator, just put everything in the numerator together over it:

[x(x-3) + 1(x+2) -1(x^2-2)] / (x-3)(x+2)

Now multiply out the top and collect terms:

= [ x^2 -3x + x + 2 - x^2 + 2] / (x-3)(x+2)

=  [-2x + 4] / (x-3)(x+2).  

You can leave it like this, or you can factor a -2 out of the top and write it as :

    [ -2(x-2)] / (x-3)(x+2).

I hope you were able to follow this, and I hope it helps you out.  Good luck.

Steve Holleran