Unformatted text preview: 2 +5 ² 1+1 = 7 and therefore P 1 does not hold. 2.1 Let r = p 3, then r 2 ± 3 = 0. Suppose now that r is rational and therefore is equal to p q for some integers p and q without any common factors. By the Rational Zeros Theorem we can now conclude that p divides 3 and q divides 1. Hence r has to be one of the following numbers ´ 1 ; ´ 3. But these numbers obviously do not satisfy the equation x 2 ± 3 = 0 and therefore cannot be r , which is a contradiction. Consequently, p 3 is not rational. 2.3 Use the Rational Zeros Theorem again or notice that if r = (2 + p 2) 1 2 is rational, then p 2 = r 2 ± 2 would be rational as well, which we know is not the case. 1...
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This note was uploaded on 02/12/2012 for the course MATH 131a taught by Professor Hitrik during the Spring '08 term at UCLA.
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