Polynomials and Equations

CHAPTER SEVEN: THE DERIVATIVE

CHAPTER SEVEN THE DERIVATIVE 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. '1.1 Differentiation Readers who are familiar with the techniques of elementary calculus will recall that for a certain type of real valued functions /(x) of one real variable, at every point e of the domain the limit 1 . I(e + h) - I(e) Im~-~~~ h-O h exists and is called the derivative oll(x) at the point e and denoted by I'(e). The function that takes e to /'(e) for every e is itself call the derivative of /(x). In geometric terms, /'(e) is the slope of the tangent to the curve y = I(x) at the point (e,/(e)). Alternatively the derivative /'(x) can be interpreted as the rate of change of the varying quantity I(x); for example, if v(t) represents the velocity of a moving body at time t, then v(t), being the rate of change of velocity, is the acceleration of the moving body at time t. The type of functions that possess a derivative include the polynomials and other elementary functions as well as many other functions. Isaac Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716) independently made use of the derivative in their separate discovery of calculus which had tremendous impact on the development of mathematics and science. 125 Polynomials and Equations In this chapter, we study the analytic properties of the derivative of a polynomial function and use them in our study of polynomials and equations. Instead of borrowing the definition of derivative from calculus, we shall start afresh by a purely algebraic approach to arrive at the same definition without using limit and convergence. Let f{x) = anxn +an_IXn- l + ... +alx +ao be a polynomial with real coefficients. Then f{x) : R -+ R is a real-valued function in one variable. For a fixed point c of the domain, every point in a neighbourhood of c can be represented by a number c + h. If we regard h as a variable quan~ity, then c+h becomes a varying point of the neighbourhood. We proceed to investigate the relation between the fixed functional value f{c) =ancn + an_ICn- l + ... + alC + ao and the varying functional value f (c +h) = an(c + ht + an- t{c + ht-l + ... + adc + h) + ao in terms of the variable quantity h. After expanding the binomials on the right-hand side of the last equality and collecting like terms that have the same exponents of h, we obtain a polynomial in the variable h: f{c +h) = Do + Dih + D2h2 + ... + Dnhn . The coefficients Do = ancn + an_ICn- 1 + ... +alc + ao DI = nanCn- 1 + {n - 1)an_ICn- 2 + ... +2c2a +al D2 = ~{n(n - 1)ancn- 2 + (n - 1)(n - 2)an_ICn- 3 + ... + 2C2} 1 Dn = - {n(n - 1)(n - 2) ... 2 . 1 . an} n! are all polynomial expressions in the fixed quantity c. Clearly the first coefficient Do is identical to f{c): which is a familiar expression. The second coefficient 126 The Derivative of the linear term Dl h can be obtained by a simple transformation of Do in which each of the n+ 1 terms arcr of Do becomes a term rarcr- 1 of D1 • Thus Dl has n terms because the last term ao of Do becomes 0 in D1 • This real number D1 , so obtained by purely algebraic means, is in fact the derivative of the function f{x) at x = c. To justify this statement, we must compare Dl with th~ analytic definition of derivative: f '( ) - r f (c + h) - f (c) c - h~ h . Substituting for f{c + h) the polynomial expression in h and taking into consideration that Do = f{c), we get f'{ c) = lim -h 1 {f(c + h) - f{c)} h-O = lim{D1 + D2h+··· + Dnhn-l} h-O Therefore the real number Dl is the derivative of f{x) at x = c which shall be denoted by f'(c) as defined in calculus...