M135W09A6 - a n-1 x n-1 + + a 1 x + a is a polynomial with...

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MATH 135 Winter 2009 Assignment #6 Due: Wednesday 4 March 2009, 8:20 a.m. Hand-In Problems 1. (a) Prove that 5 n 7 + 14 n 4 - 19 n 0 (mod 7) for all integers n . (b) Prove that 14 divides 5 n 7 + 14 n 4 - 19 n for all integers n . 2. (a) Prove by induction that if a b (mod c ) then a n b n (mod c ) for all positive integers n . (b) Prove that if p is a prime and a 2 b 2 (mod p ), then a b (mod p ) or a ≡ - b (mod p ). (c) Prove that if n is an integer, then n 2 0 (mod 4) or n 2 1 (mod 4). 3. (a) In Z 19 Fnd the inverse of [2]. (b) In Z 201 Fnd the inverse of [200]. (c) In Z 21 calculate [2] 6 + [20][2] 3 + [19]. 4. (a) ±ind the smallest positive integer k such that 8 k 1 (mod 17). (b) Let p be a prime and a be an integer not divisible by p . Let k be the smallest positive integer such that a k 1 (mod p ). Prove that k divides p - 1. (Hint: Use the division algorithm. Divide p - 1 by k ). 5. Suppose that f ( x ) = a n x n +
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Unformatted text preview: a n-1 x n-1 + + a 1 x + a is a polynomial with integer coecients. Suppose that x = c d is a rational root of f ( x ) in lowest terms. (This means that c, d Z , gcd( c, d ) = 1, and f p c d P = 0.) (a) Prove that d n a =-c ( a n c n-1 + a n-1 c n-2 d + + a 2 cd n-2 + a 1 d n-1 ). (b) Prove that c | a . (You may use the fact If g and h are integers with gcd( g, h ) = 1 and n P , then gcd( g n , h ) = 1 without proving it.) (c) Prove that d | a n . (d) ind all rational roots of the polynomial 5 x 4 + 8 x 3-4 x 2 + 7. Recommended Problems 1. Text, page 82, #5 2. Text, page 82, #7 3. Text, page 84, #58 4. Text, page 84, #61 5. Text, page 84, #65...
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