Question 1106025
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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.<br>
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.<br>
And complex roots of polynomial functions with real coefficients occur in pairs; so the minimum number of complex roots is 2.<br>
So the least possible degree of the polynomial function is 4.