math 311w homework5

math 311w homework5 - a ( x ) nor b ( x ). Problem 4. Find...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ) does not divide neither
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 nd 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...
View Full Document

Ask a homework question - tutors are online