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?

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(52818) (Show Source): You can put this solution on YOUR website!
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The graph of a rational function has a local minimum at (7,0). The complex number 4 + 2i is 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) (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.
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