Polynomials and Equations

CHAPTER ONE: POLYNOMIALS

CHAPTER ONE POLYNOMIALS 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. 1.1 Terminology We recall that a monomial in the indeterminate x is an expression of the form where a is a real number and n is a non-negative integer. The real number a is called the coefficient and the integer n is called the exponent or power of x of the monomial axB • If the coefficient is zero (a = 0) then the monomial axB is the zero monomial and is denoted simply by o. Therefore all monomials with zero coefficient are identical to the zero monomial: oxm = OxB = o. IfaxB is a non-zero monomial (a =F 0) then the exponent n is called the degree of the monomial axB • By convention the zero monomial 0 shall have no degree. Thus a monomial of degree 0 is a non-zero constant a: axo = a. It is customary to call monomials of degrees 0, 1, 2 and 3 constant, linear, quadratic and cubic monomials respectively. Expressions such as 2x3 , .Isx, 2", sin -f, e are monomials whereas expressions such as lxi, ~, sin x, eZ , log x, x + x3 are not monomials. 1 Polynomials and Equations Finally two non-zero monomials axn and bxtn in the same indeterminate x are equal if and only if they have the same coefficient and the same exponent: a = band n = ffl. Two non-zero monomials axn and bxn with the same exponent are said to be alike. For example , 0 and V2 are alike while x2 and x are unalike. Two monomials in different indeterminates, e.g. x and y, are never equal. Therefore axn :f; byrn for whatever coefficients a and b, and whatever exponents nand ffl. A polynomial in the indeterminate z is a formal sum of a finite number of unalike monomials. We usually denote polynomials in the indeterminate x by !(x), g(x), h(x), etc. By definition a monomial in x is a polynomial in x. A polynomial is usually written as in descending powers of x where the coefficient an of the first summand anxn is non-zero. The same polynomial is also written as in ascending powers of x. Either way, each summand which is a monomial is called a term of the polynomial. The numbers ao, al, ... , an in the above expressions are called the coefficients. The term ao, being a monomial of exponent 0, is called the constant term of the polynomial. The term anxn(an :f; 0) is called the leading term, its coefficient the leading coefficient and its degree the degree of the polynomial. The degree of f(x) shall be denoted by deg f(x). Thus every polynomial has a degree which is a non-negative integer except the zero polynomial which, being the zero monomial, has no degree by convention. Finally two polynomials in x are equal if and only if n = ffl and tli = bi for i = 0, 1, ... , n. It follows that in writing a polynomial as a sum of monomials it is immaterial in which order its terms appear. Two polynomials in different indeterminates are never equal. 2 Polynomials Sometimes it is very important to emphasize the fact that the coefficients CJi ofa polynomial f(x) = anxn + ... + alX + ao are real numbers. To do so, we say that f(x) is a polynomial in x with real coefficients, f(x) is a polynomial in x with coefficients in R, or !(x) is a polynomial (in x) over R. The set of all polynomials in x over R is denoted by R[x). Here the letter R indicates that the coefficients are taken from the system R of real numbers and the letter x indicates the indeterminate under consideration. We shall call R[x) the domain of polynomials in x over R or the domain of polynomials in x with coefficients in R or the domain of polynomials with real coefficients. Obviously R is a subset...