Linear Algebra, Theory And Applications, 2012a

1.6. EXERCISES 19
Eventually, for r small enough, the non constant terms are negligible and so A r is a curve
which is contained in some circle centered at a 0 which has 0 in its outside.
A r
a 0
A r r large
0
r small
Thus it is reasonable to believe that for some r during this shrinking process, the set
A r musthit0. Itfollowsthatp (z) =0forsomez. Thisisoneofthoseargumentswhich
seems all right until you think about it too much. Nevertheless, it will suffice to see that
the fundamental theorem of algebra is at least very plausible. A complete proof is in an
appendix.
1.6 Exercises
1. Let z =5+i9. Find z −1 .
2. Let z =2+i7 andletw =3− i8. Find zw,z + w, z 2 , and w/z.
3. Give the complete solution to x 4 +16=0.
4. Graph the complex cube roots of 8 in the complex plane. Do the same for the four
fourth roots of 16.
5. If z is a complex number, show there exists ω a complex number with |ω| = 1 and
ωz = |z| .
6. De Moivre’s theorem says [r (cos t + i sin t)] n = r n (cos nt + i sin nt) forn a positive
integer. Does this formula continue to hold for all integers, n, even negative integers?
Explain.
7. You already know formulas for cos (x + y) andsin(x + y) and these were used to prove
De Moivre’s theorem. Now using De Moivre’s theorem, derive a formula for sin (5x)
and one for cos (5x). Hint: Use the binomial theorem.
8. If z and w are two complex numbers and the polar form of z involves the angle θ while
the polar form of w involves the angle φ, show that in the polar form for zw the angle
involved is θ + φ. Also, show that in the polar form of a complex number, z, r = |z| .
9. Factor x 3 + 8 as a product of linear factors.
10. Write x 3 + 27 in the form (x +3) ( x 2 + ax + b ) where x 2 + ax + b cannot be factored
any more using only real numbers.
11. Completely factor x 4 + 16 as a product of linear factors.
12. Factor x 4 + 16 as the product of two quadratic polynomials each of which cannot be
factored further without using complex numbers.
13. If z,w are complex numbers prove zw = zw and then show by induction that z 1 ···z m =
z 1 ···z m . Also verify that ∑ m
k=1 z k = ∑ m
k=1 z k. In words this says the conjugate of a
product equals the product of the conjugates and the conjugate of a sum equals the
sum of the conjugates.