Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY ALGEBRA

1.8 Exercises

Polynomial equations

1.1 Continue the investigationof equation (1.7), namely

g(x)=4x^3 +3x^2 − 6 x− 1 ,

as follows.

(a) Make a table of values ofg(x) for integer values ofxbetween−2and2.Use

it and the information derived in the text to draw a graph and so determine

the roots ofg(x) = 0 as accurately as possible.

(b) Find one accurate root ofg(x) = 0 by inspection and hence determine precise

values for the other two roots.

(c) Show thatf(x)=4x^3 +3x^2 − 6 x−k= 0 has only one real root unless

− 5 ≤k≤^74.

1.2 Determine how the number of real roots of the equation

g(x)=4x^3 − 17 x^2 +10x+k=0
depends uponk. Are there any cases for which the equation has exactly two
distinct real roots?
1.3 Continue the analysis of the polynomial equation

f(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 forf′(x)

in the form 7x^5 +30x^4 +a(x−b)^2 +c, show that there is in fact only one

positive root off(x)=0.

(b) By evaluatingf(1),f(0) andf(−1), and by inspecting the form off(x)for

negative values ofx, determine what you can about the positions of the real

roots off(x)=0.

1.4 Given thatx=2isonerootof

g(x)=2x^4 +4x^3 − 9 x^2 − 11 x−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 − 2 i,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 − 10 x+8=0 areα 1 ,α 2 andα 3 then

(a) α− 11 +α− 21 +α− 31 =5/ 4 ,

(b)α^21 +α^22 +α^23 =61/ 9 ,

(c) α^31 +α^32 +α^33 =− 125 / 27.

(d) Convince yourself that eliminating (say)α 2 andα 3 from (i), (ii) and (iii) does

notgive a simple explicit way of findingα 1.

Trigonometric identities

1.7 Prove that

cos

π

12

=

√

3+1

2

√

2

by considering