Polynomial

Mathematical definition

Let $P(x)$ be a mathematical function $P:\mathbb{F} \to \mathbb{F}$

If $P(x)$, through algebraic manipulation, can be written in the form of

$$P(x)=\sum_{k=0}^{n} a_k x^k, n\in\mathbb{N},a_0, a_1, \dots, a_n\in \mathbb{F}$$

where $x\in \mathbb{F}$ is an independent variable

then $P(x)$ is a polynomial over $\mathbb{F}$. If $P(x) \neq 0$, it has degree $n$, and $a_n,a_n x^n$ are said to be its leading coefficient and leading term respectively.

$P(x)=0$ is known as the zero polynomial. Its degree is customarily defined as $-\infty$. Here's why.

$x+x^{\frac{1}{2}}$ is not a polynomial as $\frac{1}{2}\notin \mathbb{N}$

Properties

Composition of polynomials

If $P_1(x)=P_2(x)+P_3(x)$, then $\deg(P_1)=\max[\deg(P_2),\deg(P_3)]$.

If $P_1(x)=P_2(x)P_3(x)$, then $\deg(P_1)=\deg(P_2)+\deg(P_3)$

To avoid breaking this property, it is defined that $P(x)=0$ has degree $-\infty$.