EU03S-OPG

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Exercise 1: Today’s Wetware Exercise

% Solve the equation $\,(z-3)(z^2+1)=0\,.$

Exercise 2: Linear Polynomials

A polynomial $\,P:\,\mathbb C\mapsto \mathbb C\,$ is given by

$\,P(z)=(2-i)z+i\,.$

Solve the equation $\,P(z)=0\,.$

Solve the equations $\,P(z)=2\,$ and $\,P(z)=-2+2i\,.$

Exercise 3: Binomial Quadratic Equations with Real Right-Hand Sides

Let $\,r\,$ be a positive real number. Explain that the equation

$z^2=-r$

has exactly two solutions given by $\,z_0=-i\,\sqrt r\,$ and $\,z_1=i\,\sqrt r\,.$

Solve the equations $\,z^2=16\,$ and $\,z^2=-16\,.$

Solve the equations $\,z^2=17\,$ and $\,z^2=-17\,.$

Solve the equations $\,z^2=625\,$ and $\,z^2=-625\,.$

Let $\,b\,$ be a real number. Show that the solutions for $\,z^2=ib\,$ lie on line $\,y=x\,$ when $\,b>0\,$ and on the line $\,y=-x\,$ when $\,b<0\,.$

Exercise 4: Factorizations

Show (without use of the solutions formula!) that $\,\,-1+2i\,\,$ is a root in the quadratic polynomial

$\,P(z)=z^2+2z+5\,.$

Determine (without use of the solutions formula!) the second root of the polynomial and state $\,P(z)$ \, in factorized form.

Given that $\,i\,$ and $\,1+i\,$ are roots in the polynomial

$\,Q(z)=z^2-z-2iz-1+i\,\,.$

Reduce the fraction

$\frac{z^2-z-2iz-1+i}{z-1-i}\,.$

Exercise 5: The Theorem of Descent

%*Introduktion*:
%Hvis et $n$'te gradspolynomium $P$ har roden $z_0\,,$ kan $P$ omskrives således:
%
$$P(z)=(z-z_0)\cdot Q(z)$$

%hvor $Q$ er et entydigt bestemt $(n-1)$'te gradspolynomium. Men hvordan finder man $Q\,?$ Det gør man ved en divisionsalgoritme kaldet *polynomiers division* hvor $P$ divideres med førstegradspolynomiet $z-z_0\,.$ Divisionen går op, netop fordi $z_0$ er rod i polynomiet.
%Eksempel: Tredjegradspolynomiet $\,P(x)=x^3-2x^2+2x-15\,$ har roden $3\,.$ Vi dividerer roden ud således:
%\begin{center}
%\includegraphics[trim=3cm 10.3cm 3cm 16cm,width=1.0\textwidth,clip]{billeder/MohrPolynomier2.pdf}
%\end{center}
%Konklusionen er at $P$ kan skrives på faktoriseret form således:
%
$$P(z)=(z-3)\cdot (z^2+z+5)\,.$$

Show that $x_0=1$ is a root in the polynomial

$P(x)=x^3-x^2+x-1$

and determine a quadratic polynomial $Q$ such that

$P(x)=(x-1)\cdot Q(x)\,.$

Determine all complex roots for the $7$ ‘th degree polynomial

$\,P(z)=(z^6-z^5+z^4-z^3)(z-1)\,.$

State the polynomial in complete factorized form, and state the multiplicity of the roots. Hint: Use the result in the previous question.

Show that $2$ is a double root in the polynomial $\,2z^4-4z^3-16z+32\,.$

Find all solutions to the equation

$\,2z^4-4z^3-16z+32=0\,.$

%####### begin:question
%Hvilken grad har polynomiet $\,(2z^4-4z^3-16z+3)(2z^3-16)\,?$ Opskriv polynomiet på fuldstændig faktoriseret form, og angiv røddernes algebraiske multipliciteter.
%####### end:question

Exercise 6: Master the Concepts

Important clarifications concerning polynomials:

If a $n$ ‘th degree polynomial and a $n$ ‘th degree equation has equal coefficients, what is then actually the difference between them?
How many real roots has a complex polynomial of degree $n$ ?
How many real roots does a real polynomial of degree $n$ have?
How many roots both real and complex does a real polynomial of degree $n$ have?
About two $n$ ‘th degree polynomials $P$ and $Q\,$ it is known that every root in $P$ is root in $Q$ with equal algebraic multiplicity. Are the two polynomials identical?

Exercise 7: Differentiations

A real polynomial $\,P\,$ of a real variable is given by

$P(x)=25x^4-33x^3+49x^2-97x+96\,,\,\,x\in \Bbb R\,.$

Determine the derivative of $\,P\,.$

In Section $1.10$ in eNote 1 about complex numbers differentiation of complekse functions of a real variable is introduced. An example of a complex function of a real variable is the function $\,f\,$ given by

$f(t)=t-3t^3+1+i\,(t^2-5t-1)\,,\,\,t\in \Bbb R\,.$

Note that both the real part and the imaginary part of $\,f\,$ is a real polynomial of a real variable. Determine the derivative $\,f’(t)\,$ and the differential quotient $\,f’(0)\,.$

The function $\,g\,$ is given by

$g(t)=(i\cdot t^2+t-i)\cdot((1+i)t-i)\,,\,\,t\in \Bbb R\,.$

Exactly one solution to the equation $\,g’(t)=-6-i\,$ exists. Find it!

A complex polynomial $\,Q\,$ of a real variable is given by

$Q(x)=(1+2i)x^2+(2-5i)x-(1-i)\,,\,\,x\in \Bbb R\,.$

Show that $\,Q\,$ can be differentiated following the same rules that apply to real polynomials of a real variable, if only $\,i\,$ is treated as any other constant.

Exercise 8: Polynomial with Complex Coefficients

Solve the binomial Quadratic Equation $\,z^2=3-4i\,.$ Hint: use the methode in

%\tref{NUID43-ex_2gradClign}{eksempel}.

Example $2.23$ in eNote 2.

Given the polynomial

$\,P(z)=z^2-(1+2i)z-\frac 32+2i\,.$

Determine the roots of the polynomial.

Exercise 9: The Geometry of a Quadratic Equation. Enjoy!

Given the complex number $\,\,\alpha = 3-4\,i\,\,$ and the complex set of numbers

$\,S=\{\,z \in \Bbb {C}\,\,\, |\, \, \, \left|z \right| =5\,\}\,.$

Show that $\,\,\alpha \in S\,\,$ . Illustrate.

Given that the polynomial $\,\,P(z)=z^2+a\,z+b\,\,$ has real coefficients, and that $\,\alpha\,$ is a root in $\,\,P(z)\,\,$ .

State all roots of $\,\,P(z)\,\,$ , and determine $\,a\,$ and $\,b\,$ .

Let $\,c\,$ be an arbitrary real number that fulfills $\,c \in \left[-10\,;\,10\, \right] \,$ . Show that roots in the polynomial $\,\,Q(z)=z^2+c\,z+25\,\,$ belong to $\,S\,$ .