Polynomials
A basic class of functions that consists a linear combinations of monomials.
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Definition 1. Polynomials Let \(a_0,\ldots,a_n\in\mathbb{K}\). Then a real
(complex) polynomial is a function \(p:\mathbb{K}\to\mathbb{K}\) with \[p(x)=\sum_{k=0}^na_kx^k.\] If \(a_n\neq0\), then \(a_n\) is called leading
coefficient and \(n\) is called
degree of \(p\). We write
\(n=:\deg p\). If \(p\) is the zero polynomial, we set \(\deg p:=-\infty\).
The set of polynomials in \(\mathbb{K}\) is denoted by \(\mathbb{K}[x]\). Moreover, we set \[\mathbb{K}_n[x]:=\{p\in\mathbb{K}[x]\;|\;\deg
p\leq n\}.\]
Remark 2. Since sums and scalar multiples of polynomials are again polynomials, they form a vector space.
Theorem 3. Rules for the degree For \(p,q\in\mathbb{K}[x]\) holds \[\deg(p\cdot q)=\deg(p)+\deg(q),\qquad \deg(p+q)\leq\max\{\deg(p),\deg(q)\}.\]
Proof. Let \(p(x)=\sum_{k=0}^na_kx^k\), \(q(x)=\sum_{k=0}^mb_kx^k\) with \(a_n\neq0\), \(b_m\neq0\). The formula for \(\deg(p\cdot q)\) follows from \[p(x)\cdot q(x)=\sum_{k=0}^{n+m}c_kx^k\;\;\text{ for }c_k=\sum_{l=0}^ka_lb_{k-l} \ ,\] where \(a_r:=0=:b_s\) for \(r\not\in\{0,...,n\}\) and \(s\not\in\{0,...,m\}\). For the proof of the formula \(\deg(p+q)\), we assume without loss of generality that \(n\geq m\). Then \[p(x)+q(x)=\sum_{k=0}^{m}(a_k+b_k)x^k+\sum_{k=m+1}^{n}a_kx^k.\] As a consequence, we have \(\deg(p+q)\leq n=\max\{\deg(p),\deg(q)\}\). ◻
Remark 4. As the example \(p(x)=x\) and \(q(x)=-x+1\) shows, it may indeed happen
that \(\deg(p+q)<\max\{\deg(p),\deg(q)\}\).
Since \(\deg(0\cdot
p)=\deg(0)=-\infty=-\infty+\deg(p)\) and \(\deg(0+p)=\deg(p)=\max\{-\infty,\deg(p)\}\),
the choice of \(\deg p=-\infty\) makes
indeed sense for preserving the above formulas. However, this belongs to
the “not so important facts” of mathematical analysis.
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