SOLUTION: find the domain of each of the following rational functions. 2.) f(x) = 5x+6 2x-6 Simplify completely (I don't understand this) 6.) -7xy^2 28x^2 16.)

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Question 247216: find the domain of each of the following rational functions.
2.) f(x) = 5x+6
2x-6
Simplify completely (I don't understand this)
6.) -7xy^2
28x^2
16.) x-2
8x^2-32
How do you put the line between the two sets of numbers?

Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

Like this:

You write: f(x) = (5x+6)/(2x-6) [The parentheses are essential]

I read that as

The domain of a rational function is all real numbers EXCEPT any number that would make any denominator equal to 0. So what number or numbers would make ?

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There is one factor of -1 in the numerator and none in the denominator, so the -1 stays put.

There is one factor of 7 in the numerator and one factor of 7 in the denominator because 28 equals 7 times 4. So the 7 that is common to both numerator and denominator goes away, leaving you with a 4 in the denominator.

There is one factor of in the numerator and two factors of in the denominator. The one factor that is common goes away and you are left with one factor of in the denominator.

There are 2 factors of in the numerator and none in the denominator, so the 2 that are in the numerator stay put.

In summary, you have a factor of -1 and two factors of , (which is to say, one factor of ) in the numerator. You have one factor of 4 and one factor of in the denominator. Hence:

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The two terms in the denominator expression have a factor of 8 in common, so:

The binomial expression in the denominator is the difference of two squares. Use the difference of two squares factorization:

Now you can see that the numerator and denominator have a factor of in common. Eliminate that factor from both numerator and denominator:

John