. . Reciprocal Equations Let α be a solution of the equation. = .
... ( ) Then α ¹ (why?) and = . Substituting α for x in the left side of ( ), we get − + + + − + = Thus α is also a solution of ( ). Similarly we can see that if α is a solution of the equation = ...
( ) then α is also a solution of ( ). Equations ( ) and ( ) have a common property that, if we replace x by x in the equation and write it as a polynomial equation, then we get back the same equation. The immediate question that flares up in our mind is “Can we identify whether a given equation has this property or not just by seeing it?” Theorem . below answers this question.
Definition . A polynomial P x ( ) of degree n is said to be a reciprocal polynomial if one of the following conditions is true: (i) P x x P x ( ) = (ii) P x x P x ( ) = − . A polynomial P x ( ) of degree n is said to be a reciprocal polynomial of Type I if P x x P x ( ) = . is called a reciprocal equation of Type I.
A polynomial P x ( ) of degree n is said to be a reciprocal polynomial of Type II P x x P x ( ) = − . is called a reciprocal equation of Type II. - - Theory of Equations Theorem . A polynomial equation a x a x a x = , ( a n ¹ is a reciprocal equation if, and only if, one of the following two statements is true: (i) a n = , a n − = , a n − = (ii) a n =