- Prove by induction that ∑
- Prove by induction that whenever n ≥ 2,∑
- Prove by induction that 1 + ∑
- The binomial theorem states
n = ∑
Prove the binomial theorem by induction. Next show that
- Let z = 5 + i9. Find z−1.
- Let z = 2 + i7 and let w = 3 − i8. Find zw,z + w,z2, and w∕z.
- Give the complete solution to x4 + 16 = 0.
- Graph the complex cube roots of 8 in the complex plane. Do the same for the four fourth roots of 16.
- If z is a complex number, show there exists ω a complex number with = 1 and
- De Moivre’s theorem says
n = rn for
n a positive integer.
Does this formula continue to hold for all integers n, even negative integers? Explain.
- You already know formulas for cos and sin
and these were used to prove De Moivre’s
theorem. Now using De Moivre’s theorem, derive a formula for sin
and one for cos
- 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 =
- Factor x3 + 8 as a product of linear factors.
- Write x3 + 27 in the form
x2 + ax + b cannot be factored any more using
only real numbers.
- Completely factor x4 + 16 as a product of linear factors.
- Factor x4 + 16 as the product of two quadratic polynomials each of which cannot be factored further
without using complex numbers.
- If z,w are complex numbers prove zw = zw and then show by induction that ∏
j=1nzj = ∏
Also verify that ∑
k=1mzk = ∑
k=1mzk. 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
- Suppose p =
anxn + an−1xn−1 + +
a1x + a0 where all the ak are real numbers. Suppose also
that p = 0 for some
z ∈ ℂ. Show it follows that p = 0 also.
- Show that 1 + i,2 + i are the only two zeros to
so the zeros do not necessarily come in conjugate pairs if the coefficients are not real.
- I claim that 1 = −1. Here is why.
This is clearly a remarkable result but is there something wrong with it? If so, what is
- De Moivre’s theorem is really a grand thing. I plan to use it now for rational exponents, not just
Therefore, squaring both sides it follows 1 = −1 as in the previous problem. What does this tell you
about De Moivre’s theorem? Is there a profound difference between raising numbers to integer powers
and raising numbers to non integer powers?
- Review Problem 10 at this point. Now here is another question: If n is an integer, is it always true
n = cos
− isin? Explain.
- Suppose you have any polynomial in cosθ and sinθ. By this I mean an expression of the form
β=0naαβ cosαθ sinβθ where aαβ ∈ ℂ. Can this always be written in the form
m+nbγ cosγθ + ∑
n+mcτ sinτθ? Explain.
- Suppose p =
anxn + an−1xn−1 + +
a1x + a0 is a polynomial and it has n zeros,
listed according to multiplicity. (z is a root of multiplicity m if the polynomial f =
divides p but
f does not.) Show that
- Give the solutions to the following quadratic equations having real coefficients.
- x2 − 2x + 2 = 0
- 3x2 + x + 3 = 0
- x2 − 6x + 13 = 0
- x2 + 4x + 9 = 0
- 4x2 + 4x + 5 = 0
- Give the solutions to the following quadratic equations having complex coefficients. Note
how the solutions do not come in conjugate pairs as they do when the equation has real
- x2 + 2x + 1 + i = 0
- 4x2 + 4ix − 5 = 0
- 4x2 +
x + 1 + 2i = 0
- x2 − 4ix − 5 = 0
- 3x2 +
x + 3i = 0
- Prove the fundamental theorem of algebra for quadratic polynomials having coefficients in ℂ. That is,
show that an equation of the form ax2 + bx + c = 0 where a,b,c are complex numbers, a≠0 has a
complex solution. Hint: Consider the fact, noted earlier that the expressions given from the
quadratic formula do in fact serve as solutions.