A polynomial function is a 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 $a_{n}\neq 0$ .

If $p(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+...+a_{2}z^{2}+a_{1}z+a_{0}=0$ , then $z$ is called a zero of $p(x)$ . If $z$ is real, then $z$ z is a real zero of $p(x)$ ; if $z$ is imaginary, the $z$ z is a complex zero of $p(x)$ , although complex zeros include both real and imaginary 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,
Remainder theorem,
Factor theorem,
Descartes' rules of signs,
Rational zeros theorem,
Bounds on zeros theorem, a.k.a boundedness theorem
Imtermediate value theorem
Conjugate pairs theorem, and

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.