Mat246H1-Lec0101/9101

Problem Set 4

arg z + 1 =-π2，Im（z）<0。 练习2.证明以下主张
1.Foranyz∈C，z̸=1andn∈N，
1 + z + z2 +··zn = zn + 1 -1。
1 + w +··+ wn-1 = 0。 3.（奖金）如果θ∈（0,2π）和n∈N，则
1个
1 + cos（θ）+ cos（2θ）+··+ + cos（nθ）= 2 +

Exercise 1. Let z C, be such that |z| = 1 and z ̸= ±1. Show that 􏰊 z 1 􏰋 􏰄 π2 , Im (z) > 0

arg z+1 = π2, Im(z)<0. Exercise 2. Prove the following claims

1. ForanyzC,z̸=1andnN,
1+z+z2 +···zn = zn+1 1.

1+w+···+wn1 =0. 3. (Bonus)Ifθ(0,2π)andnN,then

1
1 + cos (
θ) + cos (2θ) + · · · + cos () = 2 +

Exercise 3. A natural number n is called 􏰇good􏰈 if there exists a, b Z so that a2+b2 =n.Showthatifnandmare􏰇good􏰈,thenn·misalso􏰇good􏰈.
Hint: Use complex numbers.

Exercise 4. Let P (z) = anzn + · · · + a0 be a polynomial with real coe􏰉cients. Show that if z is a root of the polynomial, then so is z ̄. Deduce that any poly- nomial with real coe􏰉cients can be written as a product of polynomials with real coe􏰉cients each one of degree 1 or 2.

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