Unit roots/Appendix

Rule 1 of chapter 2

Given $f(x)=p(x)\cdot q(x)$. Let:

$f(x)=f_0{x}^m+f_1{x}^{m-1}+\cdots +f_{m-1}x+f_m$,

$p(x)=p_0{x}^n+f_1{x}^{n-1}+\cdots +f_{m-1}x+f_m$,

$q(x)=q_0{x}^{m-n}+f_1{x}^{m-n-1}+\cdots +f_{m-n-1}x+f_{m-n}$,

(the leading coefficients $f_0$, $p_0$ and $q_0$ are nonzero). By comparing the coefficients of like terms of the expansions on both side of $f(x)=p(x)\cdot q(x)$, we get:

$f_0=p_0{q}_0$,

$f_1=p_0{q}_1+p_1{q}_0$,

$f_2=p_0{q}_2+p_1{q}_1+p_2{q}_0$,

$\dots$.

$f_{m-n}=p_0{q}_{m-n}+p_1{q}_{m-n+1}+\cdots +p_{m-n}{q}_0$,

So,

$q_0=f_0÷p_0$,

$q_1=(f_1-p_1{q}_0)÷p_0$,

$q_2=(f_2-p_1{q}_1-p_2{q}_0)÷p_0$,

$\dots$.

$q_{m-n}=(f_{m-n}-p_1{q}_{m-n+1}-\cdots -p_{m-n}{q}_0)÷p_0$.

All coefficients of $q(x)$ can be computed by the four arithmetic operations, and all division operations are division by the same nonzero number $p_0$. Now, all operation results of complex numbers are complex numbers, and all operation results of real numbers are real numbers, and all operation results of rational numbers are rational numbers. Therefore, we can conclude that if $f(x)$ and $p(x)$ have complex, real or rational coefficients, then $q(x)$ must have complex, real or rational coefficients. On the other hand, if $f(x)$ and $p(x)$ have integer coefficients, and $p_0=1$, then $q(x)$ must also have integer coefficients.