SOLUTION: I have no idea how to solve rational expressions please help. It says: SImplify the following. 1. {{{ (x+1) /(x^2-4) +3 }}} but I have no idea how to make the denominators

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I have no idea how to solve rational expressions please help. It says: SImplify the following. 1. {{{ (x+1) /(x^2-4) +3 }}} but I have no idea how to make the denominators       Log On


   

Question 716507: I have no idea how to solve rational expressions please help.
It says:
SImplify the following.
1. +%28x%2B1%29+%2F%28x%5E2-4%29+%2B3+
but I have no idea how to make the denominators the same,
Can someone explain this for me? Thank you I appreciate it!






Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
+%28x%2B1%29+%2F%28x%5E2-4%29+%2B3+
To simplify this we will need to add the terms. Since the first term is a fraction we should start by making the second term a fraction, too:
+%28x%2B1%29+%2F%28x%5E2-4%29+%2B3%2F1+
To add these fractions we will need a common denominator, preferably the lowest common denominator (LCD). The LCD is just the least common multiple (LCM) of some denominators. If you have trouble finding LCM's then it can be helpful to factor the denominators in a special way:
x^2-4  = 1 * (x+2) * (x-2)
1      = 1
LCM    = 1 * (x+2) * (x-2)
Notice how I used spacing so that the factors in each column is the same. When the factors are arranged this way, the LCM is found by multiplying 1 factor from each column. Not only does this help us find the LCM/LCD, but it also shows us which factor(s) a denominator is "missing". By comparing the first line with the LCM line we can see that that denominator not "missing" anything. So the first fraction's denominator is already the LCD. The second line is "missing" both (x+2) and (x-2). So we will multiply the second fraction by %28%28x%2B2%29%28x-2%29%29%2F%28%28x%2B2%29%28x-2%29%29:


For reasons I'll explain later I'm going to postpone multiplying out the denominator. Multiplying out the numerator:
%28%28x%2B1%29+%2F%28x%5E2-4%29%29+%2B+%283%2A%28x%5E2-4%29%29%2F%28%28x%2B2%29%28x-2%29%29+
%28%28x%2B1%29+%2F%28x%5E2-4%29%29+%2B+%283x%5E2-12%29%2F%28%28x%2B2%29%28x-2%29%29+
Now that the denominators are the same we can add the fractions (by adding the numerators):
%283x%5E2%2Bx-11%29%2F%28%28x%2B2%29%28x-2%29%29+
Next we should try to reduce this fraction. Reducing fractions involves canceling factors that are common to the numerator and denominator. So we need to factor the numerator and denominator to see if anything cancels. (This is why I left the denominator in factored form. I knew the denominators don't change while adding and I knew that I wanted it factored for when I try to reduce the fraction. Now I just have to factor the numerator.)

The numerator, however, does not factor (except to factor out a 1). So there are no factors to cancel. Now that we know the fraction does not reduce, we no longer need/want the denominator in factored form:
%283x%5E2%2Bx-11%29%2F%28x%5E2-4%29+
This is our simplified answer.

P.S. The factoring table can also be used to find greatest common factors (GCF's). While the LCM uses a factor from /i> column, the GCF only uses factors that are present in every row (not including the LCM). In the table above there is only one factor that is in both of the first two rows: 1. So the GCF is just 1. (This is why I include factors of 1 in these tables. They are useless for LCM's but they are needed sometimes for GCF's.)