The study of polynomials constitutes a major component of the mathematics course in secondary school. There polynomials first appear in connection with equations where the main concern is the evaluation of roots. Later they are treated as functions; as such we examine their derivatives, their integrals and their maxima and minima. All along we also learn the arithmetic of polynomials that involves various algebraic operations such as addition, multiplication and factorization of polynomials. In this book we shall continue to study polynomials in these three main aspects.

We recall that a monomial in the indeterminate x is an expression of...

A comparison between the number system Z and the polynomial domain R[x] will show that they are very similar as far as formal properties of addition and multiplication are concerned. In fact for most calculations that were carried out in the last chapter, we almost could have operated with polynomials as if they were integers. We shall continue to pursue this similarity in our study of the domain R[x]. In Section 1.5 we have touched upon one very special aspect of factorization of polynomials and found a necessary and sufficient condition for a linear polynomial (x − c) to be...

Equations are among the topics of mathematics that have been studied extensively for thousands of years. As equations will be the main subject for the rest of the present course, we shall begin here with a brief description of a small selection of results obtained by mathematicians in the antiquity.

In the nineteenth century archaeologists found very old Egyptian manuscripts at burial sites in the Nile valley. These manuscripts were written in ink on a kind of paper made from the papyrus plants. Among these ancient manuscripts there were books on mathematics. Of these early books on mathematics the most...

A polynomial $ g(x) = b_{m}x^{m} + b_{m-1}x^{m-1} + \cdots + b_{1}x + b_{0} $ defines a polynomial function $ g(x) = \textrm{\textbf{R}}\rightarrow \textrm{\textbf{R}} $ which maps every real number c of the domain to the real number $ g(c) $ of the range. The evaluation of $ g(x) $ at $ x = c $ is a very staight-forward matter and there are simple methods of calculation by which the correct value of $ g(c) $ can be obtained. We are now interested in the possibility of finding real values c of the domain such that $ g(c) $ coincides with an pre-assigned value d of the range. Thus given $ g(x) \in \textrm{\textbf{R}}[x] $ and $ d \in \textrm{\textbf{R}}[x] $ , we seek information on the possible values of c such that $ g(c) = d $ . In...

We remarked in the last chapter that for an equation of degree higher than four we do not possess a general method of solution and that the roots of such equations may not be obtained by root extractions and rational operations on the coefficients. Naturally this does not mean that we shall henceforth neglect the study of equations of higher degrees, but rather that we should learn individual methods to suit individual types of equations. In this chapter we pay special attention to the formal relations between the roots and the coefficients, and develop some purely algebraic methods.

We recall...

Let given be an equation

$ f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{0} = 0 $

where the coefficients have numerical real values. In our attempt to find the real roots of $ f(x) = 0 $ , it would be very advantageous if we knew the range of values in which they might occur. To put it in another way, we wish to obtain for the search of the real roots of $ f(x) = 0 $ an upper bound U so that a real number s will not be a root if $ s > U $ , and a lower bound L so that s will not be a root if $ s < L $ . For some equations such bounds can...

Up to the last chapter, only purely algebraic properties of polynomials are used in our study of equations. Beginning with this chapter, we shall put more emphasis on the functional aspect of the polynomial and examine in detail the change of the value of a polynomial corresponding to a minute increase or diminution of the variable. This will lead us to the discovery of certain basic analytic properties of polynomials such as continuity and differentiability which are usually within the purview of calculus.

Readers who are familiar with the techniques of elementary calculus will recall that for a certain type...

In the last chapter we treat polynomials as differentiable functions and study their derivatives and Taylor’s expansions. As each differentiable function is also continuous, polynomials are continuous functions. In this chapter we shall first introduce the general concept of continuous function and prove that polynomial functions are continuous. Thus every polynomial together with all its derivatives is a continuous function. Then we shall discover some very important properties of continuous functions which are useful in the theory of equations. Readers who are not familiar with the fundamental properties of real numbers and convergence may experience difficulty in reading some proofs...

The method of separation of roots based on Rolle’s theorem of the last chapter has one major disadvantage in that the roots of the equation $ f'(x) = 0 $ have to be found before the roots of $ f(x) = 0 $ can be isolated. Now if deg $ f(x) = n $ , then deg $ f'(x) = n - 1 $ . For large n, it is far more difficult to find the exact values of the real roots of the equation $ f'(x) = 0 $ than to separate the roots of $ f(x) $ by intervals. In this chapter we shall study three useful methods of separation, the best of which is discovered by the Swiss mathematician Jacques Sturm...

We recall that given an equation of degree less than five, the exact values of its roots can be written as expressions that involve only rational operations and root extractions on the coefficients. It is also known that such expressions of roots are not generally available for an equation of higher degree. Therefore, for such equations, we shall have to use numerical methods that would only give approximate decimal values to the real roots. An approximate value is always inferior to the exact value, but for many practical purposes, we only need good approximations. In this chapter we shall learn...