Polynomial function theorems for zeros

Polynomial function theorems for zeros are a set of theorems aiming to find (or determine the nature of) the polynomial remainder theorem:

• Factor theorem
• Descartes' rule of signs
• Gauss–Lucas theorem
• Rational zeros theorem
• Bounds on zeros theorem also known as the boundedness theorem
• Intermediate value theorem
• Complex conjugate root theorem
• Properties of polynomial roots

Background

A polynomial function is a function of the form

$p(x) = a_n x^n + a_{n-1} x^{n-1} + \dotsb + a_2 x^2 + a_1 x + a_0$,

where $a_i\, (i = 0, 1, 2, \dotsc, n)$ are complex numbers and $a_n \ne 0$.

If $p(z) = a_n z^n + a_{n-1} z^{n-1} + \dotsb + a_2 z^2 + a_1 z + a_0 = 0$, 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. 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. 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 do not 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).