Polynomial function theorems for zeros

Polynomial function theorems for zeros are a set of theorems aiming to find (or determine the nature) of the 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 $a_{n}\neq 0$.

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 $n\geq 1$ 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 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).