I'm reading the proof of fundamental theorem of algebra from textbook Analysis I by Amann.
I have a problem understanding the part:
Hence we can write $q$ in the form $$q=1+\alpha X^{k}+X^{k+1} r$$ for suitable $\alpha \in \mathbb{C}^{ \times}, k \in\{1, \ldots, n-1\}$ and $r \in \mathbb{C}[X]$.
I'm unable to see how we write $q$ in such form. Could you please elaborate in this issue? Thank you so much!
Best Answer
$$q(X)=1+b_1X+b_2X^2+\cdots+b_nX^n$$ with $b_n\ne0$. Maybe $b_1=0$ or even maybe $b_1=b_2=0$. In any case there is a first $k$ with $b_k\ne0$. Let $a=b_k$, so then $$q(X)=1+aX^k+b_{k+1}X^{k+1}+\cdots+b_nX^n.$$ Note that we can have $k=n$, contrary to the textbook's insistence that $k\in\{1,2,\ldots,n-1\}$. Rewrite this as $$q(X)=1+aX^k+r(X)X^{k+1}$$ where $$r(x)=b_{k+1}+b_{k+2}X+\cdots+b_n X^{n-k-1}.$$