Polynomial function theorems for zeros

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 $p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{2}x^{2}+a_{1}x+a_{0}$ where $a_{i}(i=0,1,2,...,n)$ are real and $a_{n}\neq 0$.

Found in most precalculus textbooks, these theorems include:

• The fundamental theorem of algebra,
• Descartes' rules of signs,
• Conjugate pairs theorem,
• Bounds of zeros theorem,
• Remainder theorem,
• Rational zeros theorem, and
• Imtermediate value theorem

Background

• If the degree of the polynomial function is 1, i.e., $p(x)=a_{1}x+a_{0}$, then its (only) zero is ${\frac {-a_{0}}{a_{1}}}$.
• If the degree of the polynomial function is 2, i.e., $p(x)=a_{2}x^{2}+a_{1}x+a_{0}$, then its two zeros (not necessarily distinct) are ${\frac {-a_{1}+{\sqrt {{a_{1}}^{2}-4a_{2}a_{0}}}}{2a_{2}}}$ and ${\frac {-a_{1}-{\sqrt {{a_{1}}^{2}-4a_{2}a_{0}}}}{2a_{2}}}$.

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.