4802Polynomials07

# 4802Polynomials07 - g x = n X k =0 a k x 2 k is divisible...

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Math 4802 Fall 2007 Polynomial Problems (due Tuesday, October 16) 1. Determine all polynomials P ( x ) with complex coeﬃcients such that P (0) = 0 and P ( x 2 + 1) = ( P ( x )) 2 + 1. 2. Suppose the polynomial f ( x ) = ax 3 + bx 2 + cx + d has integer coeﬃ- cients a, b, c, d with ad odd and bc even. Prove that f ( x ) has at least one irrational root. 3. Let P ( x ) be a polynomial with real coeﬃcients, and let Q ( x ) = x 3 P ( x )+ x 2 + x + 1. Prove that Q ( x ) cannot have all real roots. 4. Let n be a positive integer, and let f ( x ) be a degree n polynomial with real coeﬃcients. Prove that there are real numbers a 0 , a 1 , . . . , a n , not all zero, such that the polynomial
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Unformatted text preview: g ( x ) = n X k =0 a k x 2 k is divisible by f ( x ). 5. Do there exist polynomials a ( x ) , b ( x ) , c ( y ) , d ( y ) with complex coeﬃcients such that 1 + xy + x 2 y 2 = a ( x ) c ( y ) + b ( x ) d ( y )? 6. Prove that for any polynomial p ( x ) of degree at least 2 with integer co-eﬃcients, there exists polynomial q ( x ) with integer coeﬃcients for which the polynomial p ( q ( x )) can be factored into a product of non-constant polynomials with integer coeﬃcients. 1...
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