Polynomial: Difference between revisions
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then $P(x)$ is a '''polynomial''' | 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. | ||
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$P(x)=0$ is known as the '''zero polynomial'''. | $P(x)=0$ is known as the '''zero polynomial'''. Its degree is customarily defined as $-\infty$. Here's [[#Composition of polynomials|why]]. | ||
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== 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 $\deg(P_1)=\deg(P_2)+\deg(P_3)$ | |||
<blockquote> | |||
To avoid breaking this property, it is defined that $P(x)=0$ has degree $-\infty$. | |||
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Latest revision as of 13:44, 16 October 2026
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 $-\infty$.