SOLUTION: For the function f(x)=(x^4-625)/((x^2-25)(x+5)(x^2+25)) a) How do you find the holes in a function? b) Identify any vertical asymptotes c) Identify any holes

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Question 389522: For the function f(x)=(x^4-625)/((x^2-25)(x+5)(x^2+25))
a) How do you find the holes in a function?
b) Identify any vertical asymptotes
c) Identify any holes
Answer by robertb(4012) About Me  (Show Source):
You can put this solution on YOUR website!

a) If there are any cancellations, then the roots of the cancelled terms are candidates for the occurence of holes. The domain of f is all real numbers except -5 and 5. So it's POSSIBLE for the function to have holes there.
b) The function becomes, after cancellation, . So there is a vertical asymptote at x = -5.
c) x%5E2+-+25 was cancelled from the top and the bottom. Its roots are
-5 and 5. Since there is a vertical asymptote at x = -5, there is a "hole" at the graph only at x = 5. The expression x%5E2+%2B+25is never zero for real values of x. (The roots are imaginary.)