Unknown

1.4. Equations of Low Degree 23

(b) Show that, if f, g, h are polynomials with f = gh and f and

h are both reciprocal polynomials, then g is also a reciprocal

polynomial.

17. (a) A quartic equation of the form

x4+px3+qx2+rx+r2/p2=0

is said to be quasi-reciprocal. Show that the substitution

t = x + r/p2

leads to the equation t2 + pt + q - 2r/p = 0.

(b) A method for solving the general quartic equation can be for-

mulated as follows. Suppose the given equation can be written

in the form

x4 - qx2 - rx - s = 0.

Set x = u + v to obtain the equation

u4+4vu3+(6v2-q)u2+(4 v3-2qv-r)u+(v4-qv2-rv-s) = 0.

Show that this becomes a quasi-reciprocal equation in u if v is

chosen so that

v3 + (1/2r)(q2 + 4s)v2 + (q/2)v + (r/8) = 0.

(c) Use (a) and (b) t o o bt ain a solution to the equation

x4+3x2-2x+2=0.

Exploration

E.7. The Reciprocal Equation Substitution. The substitution t =
x + 2-l is used in solving reciprocal equations. The quantity xn + x-”
can be expressed as a polynomial pn(t) of t (see Exercise 14). Verify that
PO(t) = 2, PI(t) = t and that pn+l(t) = t p,(t)--p,-l(t) for n 2 1. Tabulate
these polynomials and look for patterns among their coefficients. Examine
the composition pm opn(t) f or indices m and n. Test the conjecture that all
coefficients of p,,(t) except the leading coefficient are divisible by n when
n is prime. Is this true? Is there any connection between the polynomials
p,(t) and the Tchebychef polynomials (Exercise 3.15)?