Exercise 4: The Binomial Theorem and Difference of Squares
Use the binomial theorem and the factorization methods for differences of squares in the two following questions, where $u$ and $v$ denote two complex numbers.
A
Reduce
$$(u+v)^2+(u-v)^2.$$
answer
$$2u^2+2v^2$$
B
Reduce
$$\frac{u^2-v^2}{u+v}+ \frac{v^2-u^2}{v-u}.$$
answer
$$2u$$
C
Compute via elementary calculations the rectangular form of the following complex numbers:
$(3+5i)(3+5i)$
$(3i+5)(3i-5)$
$\displaystyle{\frac{3-4i}{3+4i}}$
answer
$-16+30i$
$-34$
$\displaystyle{-\frac{7}{25} - \frac{24}{25} i}$
D
Prove the formula
$$z\cdot \overline{z}=\left|z\right|^2\,.$$
Exercise 5: Equations with Complex Roots
A
Solve the equation
$$(1-i)z+1=2+i\,.$$
Provide the solution on rectangular form.
answer
$$z=i$$
B
Determine all solutions to
$$(x+2i)(x-2i)(x-5)=0\,.$$
hint
What does the rule of zero product say?
answer
$$-2i \, \, , \, \, 2i \, \, , \, \, 5$$
C
Show that
$$x^4-x^3+4x^2-4x=0\,$$
has the roots $\,0,\,1,\,2i\,$ og $\,-2i\,.$
hint
A root is a solution when the right-hand-side is zero.
hint
You are given the roots. You just have to verify them one at a time.
(To be thorough, we also ought to use the Fundamental Theorem of Algebra, which says that no more than four solution exist to this equation, to clarify that we have all solutions given - this theorem will be introduced in a later week about polynomials.)
Exercise 6: Equations with Absolute Value
Find the real solutions to the following equations:
A
$$|x+3|=5$$
answer
$$x = \begin{cases} -8 \\\\ 2 \end{cases}$$
B
$$|x-2|=|3-x|$$
answer
$$x = \frac 52$$
C
What are the complex solutions to the equations above?
answer
To the former: All numbers on a circle in the complex number plane centered at $(-3,0)$ with a radius of $5$.
To the latter: All numbers with a real part of $2.5$, so:
$$x=\{2.5+ib\,\,|\,\,b\in \Bbb R\}\,.$$
Exercise 7: Sets on Roster Form
Let $A$ and $B$ be finite sets given on the following list or roster forms:
