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' rule of signs
* Rational zeros theorem
* Bounds on zeros theorem also known as the boundedness theorem
* Intermediate value theorem
* Complex conjugate root 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 numbers and $a\_n\; e\; 0$.

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 $n\; ge\; 1$ has at least one complex zero. It follows that every polynomial function of degree $n\; ge\; 1$ 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 - 4 a_2 a_0{2 a_2} and $frac\{-a\_1\; -\; sqrt$a_1}^2 - 4 a_2 a_0{2 a_2} .

A degree one polynomial is also known as a linear function, whereas a degree two 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 three polynomial) and there is a quartic formula for a quartic function (a degree four polynomial), but they are very complicated. To make matter worst, there is no general formula for a polynomial function of degree 5 or higher (see Abel–Ruffini theorem).

The theorems

Remainder theorem

The remainder theorem states that if $p(x)$ is divided by $x\; -\; c$, then the remainder is $p(c)$.
For example, when $p(x)\; =\; x^3\; +\; 2x\; -\; 3$ is divided by $x\; -\; 2$, the remainder (if we don't care about the quotient) will be $p(2)\; =\; 2^3\; +\; 2(2)\; -\; 3\; =\; 9$. When $p(x)$ is divided by $x\; +\; 1$, the remainder is $p(-1)\; =\; (-1)^3\; +\; 2(-1)\; -\; 3\; =\; -6$. However, this theorem is most useful when the remainder is 0 since it will yield a zero of $p(x)$. For example, $p(x)$ is divided by $x\; -\; 1$, the remainder is $p(1)\; =\; (1)^3\; +\; 2(1)\; -\; 3\; =\; 0$, so 1 is a zero of $p(x)$ (by the definition of zero of a polynomial function).