Simplify the expression below and explain all your steps. 4/(n^2+6n+9) - 1/(n^2-9)

Answer 1

This is no different than adding 1/2 to 1/3. You cannot add fractions without a common denominator. A common denominator has all the factors of each element in it. For example, the common denominator for 1/2 + 1/3 is 6 because 6 is 2•3.

In this case, you can't really tell what the factors are of each denominator unless you factor them.

n² + 6n + 9 = (n + 3)(n + 3)

n² - 9 = (n + 3)(n - 3)

So the common denominator needs to have two (n + 3)'s to cover the first one and an additional (n - 3) to cover the second one. So it will be (n + 3)(n + 3)(n - 3).

To achieve that common denominator for

we must multiply top and bottom by (x - 3)

To achieve the common denominator for

we must multiply top and bottom by (x + 3)

That gives us:

Then, multiply out the numerators to get (4n - 12) - (n + 3) = 4n - 12 - n - 3 = 3n - 15

So the answer is (3n - 15)/[(n + 3)(n + 3)(n - 3)]

Answer 2

According to what we know about adding and subtracting fractions with different denominators, we need to find a common denominator before we combine them. How do you think we could create a common denominator between (n^2)+6n+9 and (n^2)-9?