Reciprocal equation for IITJEE and RMO/INMO

In this set of little exercises, you will get a grip on reciprocal equations. 

reciprocal polynomial has the form

ax^{n}+bx^{n-1}+cx^{n-2}+...+cx^{2}+bx+a

in which a \neq 0 and the coefficients are symmetric about the middle one. A reciprocal equation is of the form p(t)=0 with p(t) a reciprocal polynomial.

1(a) Verify that each of the following polynomials is a reciprocal polynomial:

x^{3}+4x^{2}+4x+1

3x^{6}-7x^{5}+5x^{4}+2x^{3}+5x^{2}-7x+3

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 (x+1)q(x), with q(x) a reciprocal polynomial of even degree.

1(d) Show that, if r is a root of a reciprocal equation, then so also is 1/r.

2(a) Let ax^{2k}+bx^{2k-1}+...+rx^{k}+...+bx+a be a reciprocal equation of even degree 2k. Show that this equation can be rewritten as

a(x^{k}+x^{-k})+b(x^{k-1}+x^{-k+1})+...+r=0

2(b) Let t=x+x^{-1}. Verify that x^{2}+x^{-2}=t^{2}-2 and that x^{3}+x^{-3}=t^{3}-3t. Prove that, in general, x^{m}+x^{-m} 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 2k 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 f=gh 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

 

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