polynomial equation such that whenever α is a root, is also a root, then the polynomial equation P x ( ) = must be a reciprocal equation ” is not true. For instance is a polynomial equation whose roots are , , Note that x P x ≠ ± P ( x ) and hence it is not a reciprocal equation. Reciprocal equations are classified as Type I and Type II according to a n r − = or a n r − = − , r = , , ,... n .
We state some results without proof : • For an odd degree reciprocal equation of Type I, x = − must be a solution. • For an odd degree reciprocal equation of Type II, x = 1must be a solution. • For an even degree reciprocal equation of Type II, the middle term must be . Further x = 1and x = − are solutions.
• For an even degree reciprocal equation, by taking x + or x − as y , we can obtain a polynomial equation of degree one half of the degree of the given equation ; solving this polynomial equation, we can get the roots of the given polynomial equation. As an illustration, let us consider the polynomial equation which is an even degree reciprocal equation of Type II. So and − are two solutions of the equation and hence x − is a factor of the polynomial. Dividing the polynomial by the factor x − , we get as a factor.
Dividing this factor by x and rearranging the terms we get − + . Setting u it becomes a quadratic polynomial as u u ) − which reduces to u u . Solving we obtain u = . Taking u = gives x = and taking u = gives x =