Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY ALGEBRA1.8 ExercisesPolynomial equations1.1 Continue the investigation of equation (1.7), namelyg(x) =4x 3 +3x 2 − 6x − 1,as follows.(a) Make a table of values of g(x) for integer values of x between −2 and2.Useit and the information derived in the text to draw a graph and so determinethe roots of g(x) = 0 as accurately as possible.(b) Find one accurate root of g(x) = 0 by inspection and hence determine precisevalues for the other two roots.(c) Show that f(x) =4x 3 +3x 2 − 6x − k = 0 has only one real root unless−5 ≤ k ≤ 7 . 41.2 Determine how the number of real roots of the equationg(x) =4x 3 − 17x 2 +10x + k =0depends upon k. Are there any cases for which the equation has exactly twodistinct real roots?1.3 Continue the analysis of the polynomial equationf(x) =x 7 +5x 6 + x 4 − x 3 + x 2 − 2=0,investigated in subsection 1.1.1, as follows.(a) By writing the fifth-degree polynomial appearing in the expression for f ′ (x)in the form 7x 5 +30x 4 + a(x − b) 2 + c, show that there is in fact only onepositive root of f(x) =0.(b) By evaluating f(1), f(0) and f(−1), and by inspecting the form of f(x) fornegative values of x, determine what you can about the positions of the realroots of f(x) =0.1.4 Given that x =2isonerootofg(x) =2x 4 +4x 3 − 9x 2 − 11x − 6=0,use factorisation to determine how many real roots it has.1.5 Construct the quadratic equations that have the following pairs of roots:(a) −6, −3; (b) 0, 4; (c) 2, 2;(d)3+2i, 3 − 2i, wherei 2 = −1.1.6 Use the results of (i) equation (1.13), (ii) equation (1.12) and (iii) equation (1.14)to prove that if the roots of 3x 3 − x 2 − 10x +8=0areα 1 ,α 2 and α 3 then(a) α1 −1 + α −12+ α −13=5/4,(b) α 2 1 + α2 2 + α2 3 =61/9,(c) α 3 1 + α3 2 + α3 3 = −125/27.(d) Convince yourself that eliminating (say) α 2 and α 3 from (i), (ii) and (iii) doesnot give a simple explicit way of finding α 1 .1.7 Prove thatby consideringTrigonometric identitiescos π 12 = √3+12 √ 236