1.5 • THE ALGEBRA OF COMPLEX NUMBERS, REVISITED 3 7In Exercise 5b of Section 1.2 we asked you to show that a polynomial with
nonreal coefficients must have some roots that do not occur in complex conjugate
pairs. This last example gives an illustration of such a phenomenon.
-------.. EXERCISES FOR SECTION 1.5
- Calculate the following.
(a.) (1- iv'3)
3
(v'3+i)
2
.
( l+i).
(b) ~·(c) (v'3+i)
6
.- Show that ( v'3 + i)
4
= -8 + i8v'3
(a.) by squaring twice.
(b) by using De Moivre's formula, given in Equation (1-40).3. Use the method of Example 1.17 to establish trigonomet ric identities for cos 30
and sin30.
4. Let z be any nonzero complex number and let n be an integer. Show that zn + (z)"
is a real number.- Find all the roots in both polar and Cartesian form for each expression.
(a) (- 2 + 2i)S.(b) (-1)!.
(c) (-64)l.(d) (s)k.(e) (16i) !.
- Prove Theorem 1.5 .• the quadratic formula.
- Find all the roots of the equation z^4 - 4z^3 + 6z^2 - 4z + 5 = 0 if z 1 = i is a root
- Solve the equation (z + 1)^3 = z^3 •
- Find the three solutions to A = 4J2 + i4J2.
- Let m and n be positive integers that have no common factor. Show that there
are n distinct solutions to w" = z"' and that they are given by
Wk= r~ (cos m(^9 ~^2 dJ +isin m(^9 ~^2 dl) fork= 0, 1,... , n - 1.
11. Suppose that z :f: 1.(a) Show that 1 + z + z^2 + · · · + zn =^1 - n + l
1 :_,.