SOLUTION: The graph of a rational function has a local minimum at (7,0). The complex number 4 + 2i is ro of the function. What is the least possible degree of the function?

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Question 1106025: The graph of a rational function has a local minimum at (7,0). The complex number 4 + 2i is ro of the function. What is the least possible degree of the function?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52814) About Me  (Show Source):
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The graph of a rational function has a local minimum at (7,0). The complex number 4 + 2i is highlight%28cross%28ro%29%29 zero of the function.
What is the least possible degree of the function?
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The rational function has no "degree".

The notion, the conception of "degree" is defined for polynomial functions only.

It is NOT DEFINED for rational functions.


Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


Your question is about polynomial functions, not rational functions. A rational function is a function in the form of a fraction in which the denominator and possibly also the numerator are polynomials.

For a polynomial function to have a local minimum on the x-axis, at (7,0), it must have a root of even multiplicity at x=7. Since you are looking for the least possible degree of the function, we want it to have a double root at x=7.

And complex roots of polynomial functions with real coefficients occur in pairs; so the minimum number of complex roots is 2.

So the least possible degree of the polynomial function is 4.