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In order to add these (or any) fractions you must have common denominators. If the denominators are not the same you must make them the same before you add. It is much easier, especially with fractions of polynomials like these, to use the Lowest Commmon Denominator (LCD). To find the LCD you first factor the denominators. So we start by factoring each denominator.
The first denominator is a difference of squares and will factor according to the pattern . The second denominator is already factored and there is not much factoring that can be done with the third denominator. So let's factor the first denominator and see what we have:
At this point it looks like we have 3 different factors among the denominators: (y+3), (y-3) and (3-y). But if we can reduce the number of factors it will make things easier. And we can reduce the number of factors if we recognize that (y-3) and (3-y) are negatives of each other. So if we multiply the numerator and denominator of the third fraction by -1 then we will have just two factors among the denominators:
Now, with just the two factors, the LCD is the product of these factors (using the higher exponent for each common factor). So the LCD is:
Next we multiply the numerator and denominator of each fraction by whatever factors of the LCD its denominator is missing, if any:
We will multiply the numerators but we will leave the denominators factored for now:
With the denominators the same we can now add:
And finally, as usual with answers that are fractions, we reduce the fraction if we can. And to reduce any fraction we have to find common factors. So we need the numerator and denominator factored. The denominator is already factored. (This is why we did not multiply it out earlier!) So we just have to factor the numerator:
The numerator does not factor any more than this. And we can see that there are no common factors to cancel. So we cannot reduce this fraction. So the answer is either the fraction above or the multiplied out version of the fraction above (depending on which form is preferred):
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