## Online Course Discussion Forum

### MC III-B Help

Hi, I need help with 28, 29, 30,33,32, and 35

I meant MC III, not III-B

I will give you some hints.  If you include some details on what you have done so far, it would be better.

2.28: Use the Factor Theorem.  Plug in $x=1$ and $x=$ the imaginary cube root of unity.

2.29: Expand everything and clear the denominator, and see where that leads you,

2.30: This polynomial looks similar to a binomial expansion of degree 4.

2.32: The two sums can be combined into one, using one of them as the real part and the other as the imaginary part to form a sum of complex numbers.

2.33: Consider the binomial expansion of $(1+x)^n$, then plug in $x=\omega$ where $\omega$ is the imaginary cube root of unity.  We have the property $\omega^3=1$ and $\omega^2 + \omega + 1=0$.

2.35(a): This product can be derived from expressing the polynomial of $z^{2n}-1$ as a product of linear factors, and each factor represent one of its roots. The part (b) is similar.  Keep in mind that the roots of unity can be paired up, each pair containing two conjugate complex numbers.

For 2.30, I have

(x-i)4  = 1+i

=  $$\sqrt{2} \cdot e^{ \pi i/4}$$

But I don't know where to go from there.

For 2.32, I have

C = sum of the cosines, and S = sum of the sines.

C + iS = $$\sum{k \cdot e^{2k \pi i/n}}$$ from 1 to n-1 and I am stuck.

For 2.35, I can't see how you write it as a product of linear factors of $$z^{2n}-1$$.

See my answer to the following post for some of the details:

$$X = 1 + 2r + 3r^2 + 4r^3 + \cdots + nr^{n-1}$$
Hint: Write out $rX$, and calculate $X - rX$, and see what you get.