Chapter 3 Polynomials, operators and matrices

3.1 Polynomials

Recall from Algebra 1A (§3.2):

  • 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 \(\F [x]\).

When \(\deg p=n\), we usually write

\begin{equation*} p=a_0+a_1x+\dots +a_nx^n. \end{equation*}

Thus we adopt the convention \(x^0=1, x^{1}=x\). Here \(a_nx^n\) is the leading term of \(p\) and \(a_n\) the leading coefficient.

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

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

We can add and multiply polynomials: if

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

then

\begin{align*} p+q&:=\sum _{k=0}^{\infty }(a_k+b_k)x^k\\ pq&:=\sum _{k=0}^{\infty }(\sum _{i+j=k}a_ib_j)x^k. \end{align*} In particular, we multiply polynomials using \(x^ix^j=x^{i+j}\) and collecting terms.

The usual rules of multiplication and addition apply (in the language of Algebra 2B, \(\F [x]\) is a ring) and, in particular, \(\F [x]\) is a vector space. Moreover we have:

\begin{align*} \deg (pq)&=\deg p+\deg q,\\ \deg (p+q)&\leq \max \set {\deg p,\deg q}. \end{align*}

We can evaluate polynomials at elements of \(\F \). For \(p=a_0+\dots +a_nx^n\) and \(t\in \F \), define \(p(t)\in \F \) by

\begin{equation*} p(t):=a_0+a_1t+\dots +a_nt^{n}, \end{equation*}

where all the additions and multiplications take place in \(\F \). We say that \(t\in \F \) is a root of \(p\) if \(p(t)=0\in \F \).

Here are the main facts about evaluation:

  • • Evaluation preserves addition and multiplication: for fixed \(t\in \F \), we have

    \begin{align*} (p+q)(t)&=p(t)+q(t)\\ (pq)(t)&=p(t)q(t). \end{align*} In particular, \(p\mapsto p(t)\) is a linear map \(\F [x]\to \F \).

  • • Evaluation defines functions on \(\F \): each \(p\in \F [x]\) defines a function \(t\mapsto p(t):\F \to \F \).

  • Remark. What is a polynomial? We are used to thinking of them as the functions they define but this is not quite correct. Polynomials are simply lists of coefficients or, equivalently, sequences in \(\F \) that are eventually zero:

    \begin{equation*} \F [x]\cong \set {(a_0,\dots ,a_{n},0,0,\dots )}. \end{equation*}

    The role of the variable \(x\) is that of a placeholder to help keep track of things when we multiply polynomials.

    For some fields, different polynomials can define the same function. For example, with \(\F =\Z _2\), \(p=x^2+x\) and the zero polynomial both define the zero function1: \(p(t)=0\) for all \(t\in \Z _2\).

1 This is question 3 on exercise sheet 3.

We will need three crucial results from Algebra 1A:

  • 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 3.1 holds for any field \(\F \) but the next two results show that the field \(\C \) of complex numbers is special:

  • 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 \(p(t)=0\).

Together with Theorem 3.1, this yields:

  • 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 \(\lambda _i\).