- Prove by induction that ∑
_{k=1}^{n}k^{3}=n^{4}+n^{3}+n^{2}. - Prove by induction that whenever n ≥ 2,∑
_{k=1}^{n}>. - Prove by induction that 1 + ∑
_{i=1}^{n}i=! . - The binomial theorem states
^{n}= ∑_{k=0}^{n}x^{n−k}y^{k}whereProve 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,z
^{2}, and w∕z. - Give the complete solution to x
^{4}+ 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 ωz =.
- De Moivre’s theorem says
^{n}= r^{n}for n a positive integer. Does this formula continue to hold for all integers n, even negative integers? Explain. - You already know formulas for cosand sinand these were used to prove De Moivre’s theorem. Now using De Moivre’s theorem, derive a formula for sinand 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 x
^{3}+ 8 as a product of linear factors. - Write x
^{3}+ 27 in the formwhere x^{2}+ ax + b cannot be factored any more using only real numbers. - Completely factor x
^{4}+ 16 as a product of linear factors. - Factor x
^{4}+ 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=1}^{n}z_{j}= ∏_{j=1}^{n}z_{j}. Also verify that ∑_{k=1}^{m}z_{k}= ∑_{k=1}^{m}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. - Suppose p= a
_{n}x^{n}+ a_{n−1}x^{n−1}++ a_{1}x + a_{0}where all the a_{k}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 wrong?

- De Moivre’s theorem is really a grand thing. I plan to use it now for rational exponents, not just
integers.
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
that
^{n}= cos− isin? Explain. - Suppose you have any polynomial in cosθ and sinθ. By this I mean an expression of the form
∑
_{α=0}^{m}∑_{β=0}^{n}a_{αβ}cos^{α}θ sin^{β}θ where a_{αβ}∈ ℂ. Can this always be written in the form ∑_{γ=−(n+m ) }^{m+n}b_{γ}cosγθ + ∑_{τ=−(n+m ) }^{n+m}c_{τ}sinτθ? Explain. - Show that ℂ cannot be considered an ordered field. Hint: Consider i
^{2}= −1. - Suppose p= a
_{n}x^{n}+ a_{n−1}x^{n−1}++ a_{1}x + a_{0}is a polynomial and it has n zeros,listed according to multiplicity. (z is a root of multiplicity m if the polynomial f

=^{m}divides pbutfdoes not.) Show that - Give the solutions to the following quadratic equations having real coefficients.
- x
^{2}− 2x + 2 = 0 - 3x
^{2}+ x + 3 = 0 - x
^{2}− 6x + 13 = 0 - x
^{2}+ 4x + 9 = 0 - 4x
^{2}+ 4x + 5 = 0

- x
- 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
coefficients.
- x
^{2}+ 2x + 1 + i = 0 - 4x
^{2}+ 4ix − 5 = 0 - 4x
^{2}+x + 1 + 2i = 0 - x
^{2}− 4ix − 5 = 0 - 3x
^{2}+x + 3i = 0

- x
- Prove the fundamental theorem of algebra for quadratic polynomials having coefficients in ℂ. That is,
show that an equation of the form ax
^{2}+ 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. - Prove the Euclidean algorithm: If m,n are positive integers, then there exist integers q,r ≥ 0 such
that r < m and
Hint: You might try considering

and picking the smallest integer in S or something like this. It was done in the chapter, but go through it yourself.

- Verify DeMorgan’s laws,
where C consists of a set whose elements are subsets of a given set S. Hint: This says the complement of a union is the intersection of the complements and the complement of an intersection is the union of the complements. You need to show each set on either side of the equation is a subset of the other side.

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