Adding and Subtracting Polynomial Fractions
Purplemath
The procedure for adding numerical fractions works perfectly well on rational expressions, too; namely, you find the lcm of the (polynomial) denominators, convert to the common denominator, add the numerators, and see if there's any simplification that you can do.
For instance:
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Simplify
First, I'll find the lcm of the two denominators:
x - 4 : (x - 4) x + 4 : (x + 4) ------:---------------- LCM : (x - 4)*(x + 4)
Now I'll convert the two (polynomial) fractions to their common denominator, add, and then simplify:
![simplification: 3(3x + 4)/[(x - 4)(x + 4)]](fraction/frac47.gif)
There are no common factors between the top and the bottom, so my final answer is:
As you can see from the above example, even if your calculator can do numerical fractions for you, you will still need to know the common-denominator algorithm (that is, the process for finding and converting to common denominators) because, when you get to rational expressions (polynomial fractions), your calculator may not be able to help you — especially if you have to "show your work".
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Simplify
To find the lcm here, I'll first have to factor the two denominators.
x^2 + 5x + 6 : (x + 2)(x + 3)
x^2 + 3x + 2 : (x + 1)(x + 2)
-------------:----------------------
LCM : (x + 1)(x + 2)(x + 3)
Each of the denominators is missing one of the factors from the lcm. I'll need to multiply each fraction, top and bottom, by its missing factor. Then I'll combine the resulting fractions, and see if I can simplify at all.
![simplification: (x - 3)/[(x + 1)(x + 3)]](fraction/frac50.gif)
There was a common factor, the x + 2, shared between the numerator and the denominator, so I was able to simplify. Then my final answer is:
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