test5-fall92

# test5-fall92 - Let Î = e 2 Ï€i 5(i SHOW that 1 Î Î 2 Î 3 Î...

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Name: S.S.N: MAS 3300 – Test 5 Fall 1992 In giving proofs make sure your reasoning is clear. The questions have equal value. Write on ONE side of your paper. 1. FIND the sum, product and quotient of 3 + 2 i and 5 + i . 2. Let z = 1 - 3 i . WRITE z in polar form. 3. PROVE If z 1 , z 2 C then z 1 z 2 = z 1 z 2 . 4. PROVE that no order relation can be placed on C so that C becomes an ordered ﬁeld. 5. FIND all solutions in C to x 3 - 8 i = 0 . WRITE your solutions in rectangular form and simplify . EXHIBIT your solutions geometrically. 6. PROVE that cos72 o = cos 2 π 5 = 5 - 1 4 as follows: NOTE. You may attempt as many parts as you can.
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Unformatted text preview: Let Î¶ = e 2 Ï€i/ 5 . (i) SHOW that 1 + Î¶ + Î¶ 2 + Î¶ 3 + Î¶ 4 = 0. HINT. Expand (1-Î¶ )(1 + Î¶ + Î¶ 2 + Î¶ 3 + Î¶ 4 ), and use the fact that Î¶ 5 = 1. (ii) SHOW that Î¶ + Î¶ 4 = 2cos 2 Ï€ 5 . HINT. Remember that cos(2 Ï€-Î¸ ) = cos( Î¸ ) and sin(2 Ï€-Î¸ ) =-sin( Î¸ ). (iii) SHOW that Î¶ 2 + Î¶ 3 = ( Î¶ + Î¶ 4 ) 2-2. (iv) Hence SHOW that x = 2cos 2 Ï€ 5 is a root of the polynomial x 2 + x-1 = 0 , and that cos 2 Ï€ 5 = âˆš 5-1 4 . 1...
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