# 461hw2 - x = a/b A.5 Prove that for every positive integer...

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Math 461 F Spring 2011 Homework 2 Drew Armstrong Reading. Sections 2.1 and 2.2 Problems. A.1. Suppose that the cubic equation ax 3 + bx 2 + cx + d = 0 has three roots, called r,s,t . Give a formula for rs + rt + st in terms of a,b,c,d . A.2. Find all complex solutions z C to the quadratic equation z 2 - z + ± 1 4 - i 2 ² = 0 . A.3. Use de Moivre’s formula and the fact that cos 2 α + sin 2 α = 1 for all α R to come up with a formula for cos( θ/ 2) in terms of cos θ alone. (You can assume cos( θ/ 2) 0.) Use your formula to ﬁnd the exact value of cos( π/ 8). A.4. Let ω = cos(2 π/ 3) + i sin(2 π/ 3). Prove that for any a,b we have a 3 - b 3 = ( a - b )( a - ωb )( a - ω 2 b ) . Can you ﬁnd a similar formula for the diﬀerence a n - b n of n th powers? Hint: Factor x n - 1 and then put
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Unformatted text preview: x = a/b . A.5. Prove that for every positive integer n > 1 we have n X k =1 cos 2 πk n = 0 . Hint: Consider the number ω = cos(2 π/n ) + i sin(2 π/n ). A.6. Deﬁne a function f : C → M 2 × 2 ( R ) from the complex numbers to the 2 × 2 real matrices by setting f ( a + ib ) = ± a-b b a ² . For any complex numbers z,w ∈ C verify the following: (a) f ( z + w ) = f ( z ) + f ( w ), (b) f ( zw ) = f ( z ) f ( w ), (c) | z | 2 = det f ( z ). (The operations on the right hand sides of the equations are matrix addition, matrix multiplication, and matrix determinant.)...
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