SOLUTION: can you help me explain ? Explain what a "hole" is, and when one occurs. Give an example of a rational function that has a hole at x = - 57. thank you so much.

Algebra ->  Rational-functions -> SOLUTION: can you help me explain ? Explain what a "hole" is, and when one occurs. Give an example of a rational function that has a hole at x = - 57. thank you so much.       Log On


   

Question 597013: can you help me explain ?
Explain what a "hole" is, and when one occurs. Give an example of a rational function that has a hole at x = - 57.
thank you so much.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
A "hole" occurs in a rational function when the numerator and denominator both have a factor that could be zero. The "hole" occurs at x values that make these factors zero.

For example:
y+=+%282x%5E2+%2B+2x%29%2F%285x%2B5%29
Factoring we get:
y+=+%282x%28x+%2B+1%29%29%2F%285%28x%2B1%29%29
The numerator both have a factor of (x+1) and (x+1) could be zero so there will be a hole in the graph. The hole will occur at the value(s) of x that make (x+1) zero. That would be x = -1. So the hole in this graph will be at x = -1.

Remembering not to forget the hole, we can now cancel the common factor leaving:
y+=+%282x%29%2F5
or
y+=+%282%2F5%29x
This is the equation of a line with slope of 2/5 and a y-intercept ... and a hole at x = -1!

On the other hand,
y+=+%282x%5E3+%2B+2x%29%2F%285x%5E2%2B5%29
factors into
y+=+%282x%28x%5E2+%2B+1%29%29%2F%285%28x%5E2%2B1%29%29
Again we have a common factor. But this time it cannot be zero!. So there will not be a hole. We can still reduce the fraction:
y+=+%282%2F5%29x
The graph of this is the same line as above except that there is no hole.

To manufacture a rational function with a given hole, just start with a numerator and denominator that will be zero at the given value: (x - holenumber), For a hole at x = -57 we use (x - (-57)) or just (x + 57):
y+=+%28x%2B57%29%2F%28x%2B57%29
and then multiply the numerator and denominator by different expressions. These different expressions can be whatever you like (except a zero). Since you should multiply out the numerator and denominator I would recommend fairly simple expressions. Now we have:
y+=+%28%28expression1%29%2A%28x%2B57%29%29%2F%28%28expression2%29%2A%28x%2B57%29%29
where "expression1" and "expression2" are the two different expressions you've chosen. Multiply out the the numerator and denominator and you're finished.

If you want multiple holes, just create multiple factors. For example if you want holes at -57 and 7 you would use: