Unformatted text preview: a n-1 x n-1 + Â· Â· Â· + a 1 x + a is a polynomial with integer coeÂ²cients. Suppose that x = c d is a rational root of f ( x ) in lowest terms. (This means that c, d â 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 + Â· Â· Â· + a 2 cd n-2 + a 1 d n-1 ). (b) Prove that c | a . (You may use the fact âIf g and h are integers with gcd( g, h ) = 1 and n â P , then gcd( g n , h ) = 1â without proving it.) (c) Prove that d | a n . (d) Â±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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- Winter '08
- Integers, Prime number, Greatest common divisor, smallest positive integer