10.3 Descartes’ Rule of Signs (a) If c. C2, …, Cm are any m nonzero real numbers, and if 2 consecutive terms of this sequence have opposite signs, we say that these 2 terms present a variation of sign. With this concept, we may state Descartes’ rule of signs, a proof of which may be found in any textbook on the theory of equations, as follows: Let f(x) = 0 be a polynomial equation with real coefficients and arranged in descending powers of x. The number of positive roots of the equation is either equal to the number of variations of signs presented by the coefficients of f(x), or less than this number of variations by a positive even number. The number of negative roots is either equal to the number of variations of signs presented by the coefficients of f(-x), or less than this number of variations by a positive even number. A root of multiplicity mis counted as m roots. Investigate the nature of the roots of the following equations by means of Descartes’ rule of signs: 1. x + 3×8 5×3 + 4x + 6 = 0, 2. 2.×7 3×4 x3 – 5 = 0, 3. 3×4 + 10×2 + 5x – 4 = 0. (b) Show that xn – 1 = 0 has exactly 2 real roots if n is even, and only 1 real root if n is odd. (c) Show that x5 + x2 + 1 = 0 has 4 imaginary roots. (d) Prove that if p and q are real and q = 0, the equation x3 + px + q = 0 has 2 imaginary roots when p is positive. (e) Prove that if the roots of a polynomial equation are all positive, the signs of the coefficients are alternately positive and negative.
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