APEX Calculus

Let $p(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$ be a polynomial. If $p(a)=0\text{,}$ then $a$ is a $zero$ of the polynomial and a solution of the equation $p(x)=0\text{.}$ Furthermore, $(x-a)$ is a $factor$ of the polynomial.

An $n$th degree polynomial has $n$ (not necessarily distinct) zeros. Although all of these zeros may be imaginary, a real polynomial of odd degree must have at least one real zero.

If $p(x)=a{x}^2+bx+c\text{,}$ and $0\le b^2-4ac\text{,}$ then the real zeros of $p$ are $x=(-b\pm \sqrt{b^2-4ac})/2a$

If $p(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$ has integer coefficients, then every $rational$ $zero$ of $p$ is of the form $x=r/s\text{,}$ where $r$ is a factor of $a_0$ and $s$ is a factor of $a_n\text{.}$