To find the maximum and minimum of polynomial functions without using calculus, completing the square is effective for quadratics, while special techniques can apply to other polynomials. Knowing the zeros and their multiplicities helps identify potential extrema, as max and min points occur between the zeros. A double zero indicates that the point is either a maximum or minimum based on the sign of surrounding terms. However, these methods provide limited insight compared to derivative tests. Overall, while there are alternative approaches, calculus remains the most comprehensive method for analyzing polynomial functions.