− , a n − = − , a n − = − , Proof Consider the polynomial equation P x ( ) = a x a x a x = . … ( ) Replacing x by x in ( ), we get P x = a = . … ( ) Multiplying both sides of ( ) by x n , we get x P x = a x a x a x = . … ( ) Now, ( ) is a reciprocal equation ⇔ P x x P x ( ) = ± ⇔ ( ) and ( ) are same .
This is possible ⇔ a Let the proportion be equal to λ . Then, we get a = λ and a a n = λ . Multiplying these equations, we get λ = . So, we get two cases λ = 1and λ = − .
Case (i) : λ = In this case, we have a . That is, the coefficients of ( ) from the beginning are equal to the coefficients from the end. Case (ii) : λ = − In this case, we have a . That is, the coefficients of ( ) from the beginning are equal in magnitude to the coefficients from the end, but opposite in sign.
Note Reciprocal equations of Type I correspond to those in which the coefficients from the beginning are equal to the coefficients from the end. For instance, the equation is of type I. Reciprocal equations of Type II correspond to those in which the coefficients from the beginning are equal in magnitude to the coefficients from the end, but opposite in sign. For instance, the equation is of Type II.
Remark (i) A reciprocal equation cannot have as a solution. (ii) The coefficients and the solutions are not restricted to be real. - - (iii) The statement “ If P x ( ) = is a