document.write( "Question 597013: can you help me explain ?
\n" ); document.write( " Explain what a \"hole\" is, and when one occurs. Give an example of a rational function that has a hole at x = - 57.
\n" ); document.write( "thank you so much.
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Algebra.Com's Answer #378018 by jsmallt9(3758)\"\" \"About 
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.

\n" ); document.write( "For example:
\n" ); document.write( "\"y+=+%282x%5E2+%2B+2x%29%2F%285x%2B5%29\"
\n" ); document.write( "Factoring we get:
\n" ); document.write( "\"y+=+%282x%28x+%2B+1%29%29%2F%285%28x%2B1%29%29\"
\n" ); document.write( "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.

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

\n" ); document.write( "On the other hand,
\n" ); document.write( "\"y+=+%282x%5E3+%2B+2x%29%2F%285x%5E2%2B5%29\"
\n" ); document.write( "factors into
\n" ); document.write( "\"y+=+%282x%28x%5E2+%2B+1%29%29%2F%285%28x%5E2%2B1%29%29\"
\n" ); document.write( "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:
\n" ); document.write( "\"y+=+%282%2F5%29x\"
\n" ); document.write( "The graph of this is the same line as above except that there is no hole.

\n" ); document.write( "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):
\n" ); document.write( "\"y+=+%28x%2B57%29%2F%28x%2B57%29\"
\n" ); document.write( "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:
\n" ); document.write( "\"y+=+%28%28expression1%29%2A%28x%2B57%29%29%2F%28%28expression2%29%2A%28x%2B57%29%29\"
\n" ); document.write( "where \"expression1\" and \"expression2\" are the two different expressions you've chosen. Multiply out the the numerator and denominator and you're finished.

\n" ); document.write( "If you want multiple holes, just create multiple factors. For example if you want holes at -57 and 7 you would use:
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