Mathematics Hothouse

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.

Please do post your comments, solutions…you can take a snapshot in your smartphone, and just post it 🙂

More later…

Nalin Pithwa