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.

Found in most precalculus textbooks, these theorems include:

• Remainder theorem,
• Factor theorem,
• Descartes' rules of signs,
• Rational zeros theorem,
• Bounds on zeros theorem, a.k.a boundedness theorem
• Imtermediate value theorem, and
• Conjugate pairs theorem.

Background

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 complex 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 $z$ is a real zero of $p(x)$; if $z$ is imaginary, the $z$ is a complex zero of $p(x)$, although complex zeros include both real and imaginary zeros.

The fundamental theorem of algebra states that every polynomial function of degree $n\geq 1$ has at least one complex zero. It follows that every polynomial function of degree $n\geq 1$ has exactly $n$complex zeros, not necessarily distinct.

• 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, whereas a degree-2 polynomial is also known as a quadratic function and its two zeros are merely a direct result of the quadratic formula. However, difficulty 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 complicate. To make matter worst, there is no general formula for a polynomial function of degree 5 or higher.