If two numerical fractions or rational expressions have like denominators, we only need to add or subtract the numerators. Consider the following examples of addition.
Note that in both cases the denominators are the same. Therefore, to simplify the expressions, we need to add the numerators. Let's start with the numerical fractions.
There are two ways of adding and subtracting rational expressions and fractions with unlike denominators. Let's explore them one at a time.
First Method
The first method consists of multiplying and dividing each term by the denominator of the other term. This will result in like denominators. Therefore, from here, we can add or subtract the numerators.
ca±db=cdad±cdbc=cdad±bc
This method is valid for both numerical fractions and rational expressions. Consider the following examples of subtraction.
In both cases, we have unlike denominators. Let's use the method we explained above to perform the subtraction. We will start with the numerical fractions.
The second method for adding and subtracting numerical fractions or rational expressions with unlike denominators relies on multiplying all numerators and denominators by the leastcommon denominator. We will consider the same examples as before and subtract using this method.
Notice that we got the same answer as with the first method! Let's now subtract the rational expressions. Let's start by factoring the first denominator.
Therefore, the first denominator is a multiple of the second denominator. This means that the least common denominator is (x+1)(x−1). Thus, we need to find an equivalent fraction to x+1x−3 whose denominator is (x+1)(x−1).
Note that, once again, we obtained the same result as before. The advantage of using this method is that we do not need to factor the final expression.
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