Question

1. Find all rational roots of each of the following polynomials in Qir) a. 3+52+52 b204 11- +17 2 -11a +15 5 e. 2x3-72.2-35 f.2m-1 -2 m2 2

Can you help with question f. ?

Answer 1

Let p/q be a root of equation x^m + 2x^m - 1 - 2, gcd (p, q) = 1 (p/q)^m + 2 (p/q)^m - 1 - 2 = 0 p^m + 2 p^m - 1 middot q - 2q^m = 0 Since p^m + 2 p^m - 1 q is a multiple of P, then - 2q^m will also be multiple of p. Thus p/2 since gcd (p, q) = 1 p = 1, 2 Now, since 2p^m - 1 q - 2q^m is a multiple of 'q', thus p^m is a multiple of q = > q/p = > q = 1 since gcd (p, q) = 1 Hence we have two possibilities of roots 1/1 and 2/1 but 1^m + 2.1^m - 1 - 2 = 1 and 2^m + 2.2^m - 1 - 2 = 2^m + 2^m - 2 = 2.2^m - 2 - 2 (2^m - 1) plusminus 0 since m