## Tag Archives: higher algebra for RMO

### 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.