3.7.4 Equal Sums of Coefficients of Odd and Even Powers

. . Equal Sums of Coefficients of Odd and Even Powers Let P x ( ) = 0be a polynomial equation such that the sum of the coefficients of the odd powers and that of the even powers are equal. What does actually this mean?

If a is the coefficient of an odd degree in P x ( ) = , then the coefficient of the same odd degree in P ) = is − a . The coefficients of even degree terms of both P x ( ) = and P = are same. Thus the given condition implies that the sum of all coefficients of P = is zero and hence is a root of P = which says that − is a root of P x ( ) = . The rest of the problem of solving the equation is easy.

Example . Solve the equation2 We observe that the sum of the coefficients of the odd powers and that of the even powers are equal. Hence − is a root of the equation. To find other roots, we divide by x + and get x as the quotient.

Solving this we get and − as roots. Thus − are the roots or solutions of the given equation.