Advanced Algebra Study Notes (2) Polynomials - Unary Polynomials

2. Unary polynomial

Definition 2 Let n be a non-negative integer. Formal expression
$$a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0,\left(1\right)$$
a$a_0,a_1,\dots ,a_n$All belong to the number field $P$ , calledthe coefficient in the number field PPA univariate polynomial in$P$ , or simply the numberPPA univariate polynomial over$P.$ aixk
$a_i{x}^k$ is called the kthterm,$a_i$It is called the kth term coefficient . Use $f(x),g(x),...$ or$f，g，...$ to represent polynomials.

Definition 3 If the polynomial $f\left(x\right)$ and$In\; g\left(x\right)$ , except for items with zero coefficients, the coefficients of the same order items are all equal, then$f\left(x\right)$ and$g\left(x\right)$ is called equal, recorded as
$$f(x)=A\; polynomial\; whose\; g\left(x\right)$$
coefficients are all zero is called a zero polynomial, denoted as 0.

In (1), if $a_n\ne 0$ , n$a_n{x}^n$ is called the first term of the polynomial (1),$a_n$is called the leading coefficient. n is called the degree of the polynomial (1). The zero polynomial is the only polynomial with an undefined degree. The polynomial $The\; degree\; of\; f\left(x\right)$ is recorded as
$$\partial \left(f\right(x))$$
Because the zero polynomial does not define the degree, so use the symbol$\partial \left(f\right(x))$ , always assume$f(x)\ne 0$.

For the convenience of calculation and discussion

Introducing the ampersand
Let
$$\begin{gathered} f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0 \\ g(x)=b_m{x}^m+b_{m-1}{x}^{m-1}+\cdots +b_0 \end{gathered}$$
are two polynomials on the number field P, then it can be written as
$$\begin{gathered} f\left(x\right)=\sum _{i=1}^n{a}_i{x}^i \\ g\left(x\right)=\sum _{i=1}^m{b}_j{x}^j \end{gathered}$$
若m=n,则
$$\begin{array}{c}\begin{array}{rl}f(x)+g(x)& =(a_n+b_n)x^n+(a_{n-1}+b_{n-1})x^{n-1}+\cdots +(a_1+b_1)+(a_0+b_0)\\ & =\sum _{i=1}^n(a_i+b_i)x^i\end{array}\end{array}$$

while $f\left(x\right)$ and$g(x)$的乘积为
$$f(x)\cdot g(x)=a_n{b}_m{x}^{n+m}+(a_n{b}_{m-1}+a_{n-1}{b}_m)x^{m+n-1}+\cdots +(a_1{b}_0+a_0{b}_1)x+a_0{b}_0$$
where $The\; coefficient\; of\; the\; s$ term is
$$a_s{b}_0+a_{s-1}{b}_1+\cdots +a_1{b}_{s-1}+a_0{b}_s=\sum _{i+j=s}{a}_i{b}_j$$
So $f\left(x\right)g\left(x\right)$ can be expressed as
$$f(x)g(x)=\sum _{s=0}^{m+n}(\sum _{i+j=s}{a}_i{b}_j)x^s$$

Obviously, after adding, subtracting, multiplying, and dividing two polynomials on the number field P, the result is still a polynomial on the number field P.

For polynomial addition and subtraction , it is not difficult to see that
$$\partial (f(x)\pm g(x))\le max(\partial (f(x)),\partial (g(x)))$$

max(n,m) represents the larger number among n and m.

For polynomial multiplication, it can be proved that if $f(x)\ne 0,g(x)\ne 0$ , then$f(x)g(x)\ne 0$,并且
$$\partial (f(x)g(x))=\partial (f(x))+\partial (g(x))$$

设
$$\begin{gathered} f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0 \\ g(x)=b_m{x}^m+b_{m-1}{x}^{m-1}+\cdots +b_0 \end{gathered}$$

where $a_n\ne 0,b_m\ne 0$ , sof ( x ) g ( x ) f(x)g(x)The first term of$f$$($$x$$)$$g$$($$x$$)$
$$a_n{b}_m{x}^{n+m}$$

Obviously $a_n{b}_m\ne 0$ , so$f(x)g(x)\ne 0$ , and its degree is n+m.
It can also be seen from the above thatthe leading coefficient of the polynomial product is equal to the product of the leading coefficients of the factors.

Laws satisfied by polynomial operations
1. Commutative law of addition
$$f(x)+g(x)=g\left(x\right)+f\left(x\right)$$

2. Associative law of addition
$$(f(x)+g(x))+h(x)=f(x)+(g(x)+h(x))$$

3. The commutative law of multiplication
$$f(x)g(x)=g(x)f(x)$$

4. Associative law of multiplication
$$(f(x)g(x))h(x)=f(x)(g(x)h(x))$$

5. The distributive law of multiplication and addition
$$f(x)(g(x)+h(x))=f(x)g(x)+f(x)h(x)$$

6. Multiplicative elimination law
If $f(x)g(x)=f(x)h(x)$且$f(x)\ne 0$，那么
$$g(x)=h(x)$$

Definition 4 The whole of unary polynomials with all coefficients in the number field P is called a closed loop of unary polynomials in the number field P, denoted as P[x], and P is called the coefficient field of p[x].