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Polynomial function theorems for zeros

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Polynomial function theorems for zeros are a set of theorems aiming to find (or determine the nature) of the complex zeros of a polynomial function of the form where are real and .

Found in most precalculus textbooks, these theorems include:

Background

  • If the degree of the polynomial function is 1, i.e., , then its (only) zero is .
  • If the degree of the polynomial function is 2, i.e., , then its two zeros (not necessarily distinct) are and .

A degree-1 polynomial is also known as a linear function where a degree-2 polynomial is also known as a quadratic function and the two zeros are merely a direct result of the quadratic formula. However, problem rises when the degree of the polynomial, n, is higher than 2. It is true that there is a cubic formula for a cubic function (a degree-3 polynomial) and there is a quartic formula for a quartic function (a degree-4 polynomial), but they are very complicated. To make matter worst, there is a general formula for a typical polynomial function of degree 5 or higher.