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22 1. Fundamentals

(a) Verify that each of the following polynomials is a reciprocal poly-

nomial:

x3 + 4x2 + 4x + 1

3x6 - 7x5 + 5x4 + 2x3 + 5x2 - 7x + 3.

(b) Show that 0 is not a zero of any reciprocal polynomial.

(c) Show that -1 is a zero of any reciprocal polynomial of odd

degree, and deduce that any reciprocal polynomial of odd degree

can be written in the form (x + l)q(x), with q(x) a reciprocal

polynomial of even degree.

(d) Show that, if r is a root of a reciprocal equation, then so also is

l/r.

14. (a) Let QX~~ + bx2k-1 +... + rxk +... + bx + a = 0 be a recipro-
    cal equation of even degree 2k. Show that this equation can be
    rewritten

a(xk + xmk) + b(xk-’ + xv’+‘) -I-... + r = 0.

(b) Let t = x+x -l. Verify that x2+xs2 = t2-2 and that x3+xm3 =

t3-3t. Prove that, in general, x”‘+x-~ is a polynomial of degree

m in t.

(c) Use the substitution in (b) to show that the reciprocal equation

in (a) can be rewritten as an equation of degree k in the variable

t. Deduce that the solution of a reciprocal equation of degree 2k

can in general be reduced to solving one polynomial equation of

degree k as well as at most k quadratic equations.

15. (a) Show that the transformation t = x+x-’ applied to the equation

2x4+5x3+x2+5x+2=0

leads to the equation

2t2 + 5t - 3 = 0.

Solve the latter equation for t and use the result to obtain solu-

tions to the original equation.

(b) As a check, verify that the left side of the equation in x can be

written as the product of the two quadratic polynomials which

arise in solving for x once the two values oft are found.

16. (a) Show that a product of reciprocal polynomials is a reciprocal
    polynomial.