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math 311w homework5

# math 311w homework5 - a x nor b x Problem 4 Find a d x =...

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Homework 5. Due Friday, February 19, 2009 Give rigorous solutions of all problems. Problem 1. Prove that a · b = gcd ( a, b ) lcm ( a, b ) Hint: use the fundamental theorem of arithmetics . Problem 2. Prove that 6 is an irrational number. Hint: use the fundamental theorem of arithmetics. Problem 3. a) Prove by induction that if a product of n polynomials is divisible by an irreducible polynomial p ( x ) then at least one of them is divisible by p ( x ). You can assume without a proof that this fact is true for two polynomials. b) Give an example of three polynomials a ( x ) , b ( x ) and c ( x ), such that c ( x ) divides a ( x ) · b ( x ), but c ( x
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Unformatted text preview: a ( x ) nor b ( x ). Problem 4. Find a d ( x ) = gcd ( a ( x ) ,b ( x )) and write it as polynomial linear combination of a ( x ) and b ( x ): a) a ( x ) = x 4 + 1 and b ( x ) = x 3 + x 2 + x + 1. b) a ( x ) = x 2 + 3 x + 2 and b ( x ) = x 3 + x 2 + x + 1. Problem 5. For the equations below either ﬁnd a solution or explain why they do not have a solution: a) x 2 + 5 x + 4 = n ( x )( x 2 + 3 x + 2) + m ( x )( x 3 + x 2 + x + 1) b) x 2 + 4 = n ( x )( x 2 + 3 x + 2) + m ( x )( x 3 + x 2 + x + 1) 1...
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