Math Rational Function

Question

Find an equation for the rational function which has a zero at x = 5, a hole at x = 4, and asymptotes at x = 7 and x = 6?

 

What is the best way to answer and check this ?

Answer 1

 

The hole of a rational function occurs when a value produces 0 in the numerator and denominator of the function.

This is more easily seen in factored form

 

In order to have a hole at  x = 4,  (x-4) must be a factor of the numerator and denominnator.  Recall that certical asymtotes occur at valus that creat division by 0.

 

In order to have vertical assymptotes at x = 7 and x = 6,  (x -7) and (x -6) must be factors of the denominator.

 

When we merge these together, we obtain the following "factored" rational function

 

(x - 4)(x - 5)

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(x - 4)(x - 7)(x - 6)

 

We can multiply the terms to get polynomial expressions, if desired.

 

x² - 9x + 20

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x³ -17x² + 94x - 168