Chapter 3 Polynomials, operators and matrices

3.1 Polynomials

  • Definitions. A polynomial in a variable \(x\) with coefficients in a field \(\F \) is a formal expression

    \begin{equation*} p=\sum _{k=0}^{\infty }a_kx^k \end{equation*}

    with coefficients \(a_k\in \F \) such that only finitely many \(a_k\) are non-zero.

    Two polynomials are equal if all their coefficients are equal.

    The zero polynomial has all coefficients zero.

    The degree of a polynomial \(p\) is \(\deg p=\max \set {k\in \N \st a_k\neq 0}\). By convention, \(\deg 0=-\infty \).

    The set of all polynomials in \(x\) with coefficients in \(\F \) is denoted \(\F [x]\).

  • Definition. A polynomial is monic if its leading coefficient is \(1\):

    \begin{equation*} p=a_0+\dots +x^{n}. \end{equation*}

  • Theorem 3.1 (Algebra 1A, Proposition 3.10). Let \(p,q\in \F [x]\). Then there are unique \(r,s\in \F [x]\) such that

    \begin{equation*} p=sq+r \end{equation*}

    with \(\deg r<\deg q\).

  • Theorem 3.2 (Fundamental Theorem of Algebra). Let \(p\in \C [x]\) be a polynomial with \(\deg p\geq 1\). Then \(p\) has a root. Thus there is \(t\in \C \) with \(p(t)=0\).

  • Theorem 3.3. Let \(p\in \C [x]\) and \(\lst \lambda 1k\) the distinct roots of \(p\). Then

    \begin{equation*} p=a\prod _{i=1}^k(x-\lambda _i)^{n_i}, \end{equation*}

    for some \(a\in \C \) and \(n_i\in \Z _+\), \(1\leq i\leq k\).

    \(n_i\) is called the multiplicity of the root \(\lambda _i\).