In this set of little exercises, you will get a grip on reciprocal equations.
A reciprocal polynomial has the form
in which and the coefficients are symmetric about the middle one. A reciprocal equation is of the form
with
a reciprocal polynomial.
1(a) Verify that each of the following polynomials is a reciprocal polynomial:
1(b) Show that 0 is not a zero of any reciprocal polynomial.
1(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 , with
a reciprocal polynomial of even degree.
1(d) Show that, if r is a root of a reciprocal equation, then so also is .
2(a) Let be a reciprocal equation of even degree
. Show that this equation can be rewritten as
2(b) Let . Verify that
and that
. Prove that, in general,
is a polynomial of degree m in t.
2(c) Use the substitution in 2b to show that the reciprocal equation in 2a can be rewritten as an equation of degree k in the variable t. Deduce that the solution of a reciprocal equation of degree can in general be reduced to solving one polynomial equation of degree k as well as at most k quadratic equations.
3(a) Show that a product of reciprocal polynomials is a reciprocal polynomial.
3(b) Show that, if f, g, h are polynomials with and f and h are both reciprocal polynomials, then g is also a reciprocal polynomial.
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More later…
Nalin Pithwa