Example 2 Find the product of Solution First, we divide the numerator and denominator by the common factors to get Now, multiplying the remaining factors of the numerators and denominators yields If a negative sign is attached to any of the factors, it is advisable to proceed as if all the factors were positive and then attach the appropriate sign to the result.
A positive sign is attached if there are no negative signs or an even number of negative signs on the factors; a negative sign is attached if there is an odd number of negative signs on the factors.
Thus, We subtract fractions with unlike denominators in a similar way that we add such fractions. We build each fraction to an equivalent fraction with this denominator to get Now, adding numerators yields Again, special care must be taken with binomial numerators.
However, we first write each fraction in standard form. Example 2 Write the difference of as a single term.
We now build each fraction to fractions with this denominator and get We can now add the numerators, simplify, and obtain Common Errors Note that we can only add fractions with like denominators.
Thus, Also, we only add the numerators of fractions with like denominators.
In general, Example 1 When subtraction is involved, it is helpful to change to standard form before adding.
Example 2 We must be especially careful with binomial numerators.
To find the LCD: Example 1 Find the lowest common denominator of the fractions Solution The lowest common denominator for contains among its factors the factors of 12, 10, and 6. (This number is the smallest natural number that is divisible by 12, 10, and 6.) The LCD of a set of algebraic fractions is the simplest algebraic expression that is a multiple of each of the denominators in the set.
Thus, the LCD of the fractions because this is the simplest expression that is a multiple of each of the denominators.