# 多項式

## 定義

$a_0 + a_1 X + \cdots + a_{n - 1} X^{n - 1} + a_n X^n$

$p_0(X_1) + p_1(X_1) X_2 + \cdots + p_{n_2 - 1}(X_1) X_2^{n_2 - 1} + p_{n_2} (X_1) X_2^{n_2}$

$-5X^3 Y$

### 多項式的升冪及降冪排列

$\ 2X^5 Y^2 + 7X^3 Y^4 + 8X^1 Y^6$

$2Y^2 X^5 + 7Y^4 X^3 + 8Y^6 X^1.$

## 多項式的運算

### 多項式的加法

\begin{align} {\color{BrickRed} P} &= {\color{BrickRed} 3X^2 - 2X + 5XY - 2} \\ {\color{RoyalBlue} Q} &= {\color{RoyalBlue} -3X^2 + 3X + 4Y^2 + 8} \end{align}

${\color{BrickRed} P} + {\color{RoyalBlue}Q} =( {\color{BrickRed} 3X^2 - 2X + 5XY - 2} )\; + \; ({\color{RoyalBlue} -3X^2 + 3X + 4Y^2 + 8})$

$P + Q = X + 5XY + 4Y^2 + 6$

### 多項式乘法

\begin{align} \color{BrickRed} P &= \color{BrickRed}{2X + 3Y + 5} \\ \color{RoyalBlue} Q &= \color{RoyalBlue}{2X + 5Y + XY + 1} \end{align}

$\begin{array}{rccrcrcrcr} {\color{BrickRed}P}{\color{RoyalBlue}Q}&{{=}}&&({\color{BrickRed}2X}\cdot{\color{RoyalBlue}2X}) &+&({\color{BrickRed}2X}\cdot{\color{RoyalBlue}5Y})&+&({\color{BrickRed}2X}\cdot {\color{RoyalBlue}XY})&+&({\color{BrickRed}2X}\cdot{\color{RoyalBlue}1}) \\&&+&({\color{BrickRed}3Y}\cdot{\color{RoyalBlue}2X})&+&({\color{BrickRed}3Y}\cdot{\color{RoyalBlue}5Y})&+&({\color{BrickRed}3Y}\cdot {\color{RoyalBlue}XY})&+& ({\color{BrickRed}3Y}\cdot{\color{RoyalBlue}1}) \\&&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}2X})&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}5Y})&+& ({\color{BrickRed}5}\cdot {\color{RoyalBlue}XY})&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}1}) \end{array}$

$PQ = 4X^2 + 21XY + 2X^2Y + 12X + 15Y^2 + 3XY^2 + 28Y + 5$

### 多項式除法

$A = BQ + R$

$\begin{matrix} \qquad\quad\;\, X^2 \; - 9X \quad - 27\\ \qquad\quad X-3\overline{\vert X^3 - 12X^2 + 0X - 42}\\ \;\; \underline{\;\;X^3 - \;\;3X^2}\\ \qquad\qquad\quad\; -9X^2 + 0X\\ \qquad\qquad\quad\; \underline{-9X^2 + 27X}\\ \qquad\qquad\qquad\qquad\qquad -27X - 42\\ \qquad\qquad\qquad\qquad\qquad \underline{-27X + 81}\\ \qquad\qquad\qquad\qquad\qquad\qquad\;\; -123 \end{matrix}$

## 因式分解

$P = (X+1)(X-1)(X^4 -X^2 + 1)$

$P = (X+1)(X-1)(X^2 -\sqrt{3} X + 1)(X^2 + \sqrt{3} X + 1),$

$P = (X+1)(X-1)(X - \frac{\sqrt{3} + i}{2})(X - \frac{\sqrt{3} - i}{2})(X + \frac{\sqrt{3} + i}{2})(X + \frac{\sqrt{3} - i}{2}).$

## 多項式函數

$f_P : \; \; \mathbb{A} \longrightarrow \mathbb{A}$
$x \; \; \mapsto a_0 + a_1 x + \cdots + a_n x^n = P(x)$

### 多項式方程

$f_P(x) = a_0 + a_1 x + \cdots + a_n x^n = 0.$

$x^3 + 3x - 4 = 0$

## 多項式的分析特性

$f_P(x) = a_0+a_1x + \cdots + a_n x^n = \sum_{k=0}^n a_k x^{k}.$

$f_P'(x) = a_1 + 2a_2 x + \cdots + n a_n x^{n-1} = \sum_{k=1}^n k a_k x^{k-1}.$

$\int f_P (x) \; \mathrm{d} x = C + a_0x + \frac12 a_1 x^2 + \cdots + \frac{1}{n+1} a_n x^{n+1} = C + \sum_{k=0}^n \frac{1}{k+1} a_k x^{k+1}.$

$\mathrm{D}( P) = a_1 + 2a_2 X + \cdots + n a_n X^{n-1} = \sum_{k=1}^n k a_k X^{k-1}.$

$\mathrm{I}(P) = a_0 X + \frac12 a_1 X^2 + \cdots + \frac{1}{n+1} a_n X^{n+1} = \sum_{k=0}^n \frac{1}{k+1} a_k X^{k+1}.$

## 參考來源

1. ^ 1.0 1.1 Edwards, Harold M. Linear Algebra. Springer. 1995: 47. ISBN 9780817637316.
2. ^ Salomon, David. Coding for Data and Computer Communications. Springer. 2006: 459. ISBN 9780387238043.