4.6 Polynomial function

It is certainly well known to the reader how to define the integer power of a real number $a$. Let us recall it here. For a natural number $n$ we set

\begin{equation}\label{eq-exp1}\tag{4.10} a^n := \underbrace{a \cdot a \cdots a}_{n\,\text{times}} \end{equation}

and for $n=0$, $a^0 := 1$ (in the context of this section, even in the case $a=0$). For negative integers $n$ and non vanishing $a$ we define $a^n := \frac{1}{a^{-n}}$. The number $-n$ is then positive so we can use it in the denominator (4.10). For examples, it holds

\begin{equation*} \pi^0 = 1, \quad 2^4 = 2\cdot 2\cdot 2 \cdot 2 = 16, \quad 3^{-2} = \frac{1}{9}, \quad 0^0 = 1. \end{equation*}

According to this definition of power, it is obvious that for every real non-zero $a$ and integer $k$ and $n$ we have the important relationships (think!)

\begin{equation}\label{eq-exp2}\tag{4.11} a^k \cdot a^n = a^{k+n} \quad \text{a} \quad \big(a^k\big)^n = a^{kn}. \end{equation}

The „power“ operation with $a>0$ can be defined not only for integer coefficients. However, at this point it is not clear how to define (let alone calculate) the value of an expression such as for example $3^\pi$ or $1.2^{2.8}$. This issue is discussed in more detail in  BIE-ZMA.

The generalizations of linear and quadratic functions are polynomials. We call polynomial function each function of the form

\begin{equation*} f(x) = \sum_{k=0}^n a_k x^k, \quad x\in D_f = \mathbb{R}. \end{equation*}

If $a_n \neq 0$, we say that $n$ is the degree. (or order) of $f$. The real constants $a_0, a_1, \ldots, a_n$ determine the function $f$ as in previous cases the constants $a, b, c$ did for the linear, resp. quadratic, function. These constants are often called polynomial coefficients. To emphasize the field in which we work, we sometimes speak of the functions introduced above as real polynomials.

Of course, polynomial functions include both linear and quadratic functions. A common feature of polynomials is that only adding and multiplying operations are needed to calculate their functional values. In this sense, they are indeed one of the simplest (elementary) functions. In addition, these operations can be cheap on CPU, resp. FPU, and therefore the evaluation of polynomial functional values is easy.

The domain of any polynomial is the whole real line, $D_f = \R$. If the degree of the polynomial is odd, then its image set is $\R$. However, if the polynomial degree is even, then only a portion of the real axis contains the image set (in particular a certain interval or a point in the case of a constant polynomial).

Finding the roots of polynomials is generally a complicated task. Explicit formulas like for example (4.9), are known only for polynomials of degree $1$, $2$, $3$ and $4$. For higher degree polynomials, not only formulas for the roots are unknown, it is also proven that they do not exist. Let us emphasize this fact once more. If a polynomial of degree at least five is given, then the formula to find its roots does not exist and will never exist. When searching for roots then we have to resort to numerical methods42.

Question 4.5

Which of the following functions is a polynomial?

  1. $f(x) = x^2 + 2x + 3 + \frac{4}{x}$,

  2. $f(x) = x\sin(2)-x^3$,

  3. $f(x) = e^{2\ln (1+x^2)}$,

  4. $f(x) = \frac{x^3 + x}{x^2+1}$

Show answer

1. no, 2. yes, 3. yes, 4. yes.

The only high school „method“ to search for the roots of a polynomial $P (x)$ of degree greater than two43 consists in repeating the following steps

  1. guess one root, let us call it $\lambda$,

  2. find (for example by the polynomial division) the root factor, i.e. factorize $P(x) = (x - \lambda) Q(x)$, where the polynomial $Q(x)$ is of one degree lower than $P(x)$,

  3. return to the first point, but now guess the root of the polynomial $Q(x)$.

This process is repeated until we reach a polynomial of the degree two, for which we can find the roots by using the formula (4.9).

It is advisable to realize that this procedure is not an algorithm solving the task of finding the roots of a given polynomial. The first task is an esoteric step leaning on randomness. For example, try applying the procedure to the following polynomial (still of quite small degree):

\begin{equation*} 12 \, x^{6} - x^{5} + 57 \, x^{4} - 10 \, x^{3} - 9738 \, x^{2} + 759 \, x + 47817. \end{equation*}

At this point we advise the reader to review the polynomial division algorithm. This algorithm finds use not only in the task of finding the roots of polynomials, but also finds very important applications in computer security and encryption.

Question 4.6

Find the roots of the following polynomials.

  1. $x^2 + x - 12$,

  2. $x^3 - 2x^2 - 5x + 6$,

  3. $x^3 + 2x^2 - 4x - 8$.

Show answer

1. $3$ a $-4$, 2. $1$, $-2$ a $3$, 3. $-2$ a $2$.