A polynomial $\,P:\,\mathbb C\mapsto \mathbb C\,$ is given by
$\,P(z)=(2-i)z+i\,.$
A
Solve the equation $\,P(z)=0\,.$
answer
$$\frac15-\frac25\,i$$
B
Solve the equations $\,P(z)=2\,$ and $\,P(z)=-2+2i\,.$
answer
$$1 \quad\text{ and }\quad -1$$
Exercise 3: Binomial Quadratic Equations with Real Right-Hand-Sides
A
Let $\,r\,$ be a positive real number. Explain why the equation
$$z^2=-r$$
has exactly two solutions given by $\,z_0=-i\,\sqrt r\,$ and $\,z_1=i\,\sqrt r\,.$
B
Solve the equations $\,z^2=16\,$ and $\,z^2=-16\,.$
answer
First equation: $z=4$ and $z=-4$. Second equation: $z=4i$ and $z=-4i$.
C
Solve the equations $\,z^2=17\,$ and $\,z^2=-17\,.$
answer
First equation: $\,z=\sqrt{17}\,$ and $\,z=-\sqrt{17}\,$. Second equation: $\,z=i\sqrt{17}\,$ and $\,z=-i\sqrt{17}\,$.
D
Solve the equations $\,z^2=625\,$ and $\,z^2=-625\,.$
answer
First equation: $z=25$ and $z=-25$. Second equation: $z=25i$ and $z=-25i$.
E
Let $\,b\,$ be a real number. Show that the solutions to
$\,z^2=ib\,$ lie on the line $\,y=x\,$ when $\,b>0\,$ and on the line $\,y=-x\,$ when $\,b<0\,.$
hint
The line $y=x$ is a first- and third-quadrant bisector, meaning a line tilted $45^\circ$ ($\frac{\pi}{4}$) passing through the origin.
hint
Your aim is to show that any solution $z$ has an argument of $\frac{\pi}4$ when $b>0$ regardless of the actual value of $b$. Try to equate the arguments of either side of the equation.
Exercise 4: Factorizations
A
Without using any solution formula, show that $\,\,-1+2i\,\,$ is a root of the quadratic polynomial
$$\,P(z)=z^2+2z+5\,.$$
State based on this result the second root of the polynomial and write $\,P(z)$ on factorized form.
answer
The second root is $-1-2i$.
$$P(z)=(z+1+2i)(z+1-2i)$$
B
Given that $\,i\,$ and $\,1+i\,$ are roots of the polynomial
$$\,Q(z)=z^2-z-2iz-1+i\,\,,$$
simplify the fraction
$$\frac{z^2-z-2iz-1+i}{z-1-i}\,.$$
answer
$$z-i$$
Exercise 5: The Theorem of Descent
A
Show that $x_0=1$ is a root of the polynomial
$$P(x)=x^3-x^2+x-1$$
and determine a quadratic polynomial $Q$ such that
$$P(x)=(x-1)\cdot Q(x)\,.$$
answer
$$Q(x)=x^2+1$$
B
Compute all complex roots of the 7th degree polynomial
$$\,P(z)=(z^6-z^5+z^4-z^3)(z-1)\,.$$
Write the polynomial on its fully factorized form and state the multiplicity of its roots.
hint
Use the result to the previous question. You may need to do one factorization first.
answer
Fully factorized form:
$$P(z)=z^3(z-1)^2(z-i)(z+i)$$
Roots:
$0$ with multiplicity $3$
$1$ with multiplicity $2$
$i$ with multiplicity $1$
$-i$ with multiplicity $1$
C
Show that $2$ is a double root of the polynomial $\,2z^4-4z^3-16z+32\,.$
hint
Show by substitution that it is a root. After a descension, show by substition that it is a root once again.
D
Find all solutions to the equation
$$\,2z^4-4z^3-16z+32=0\,.$$
answer
$$2,\,-1+i\sqrt 3,\, -1-i\sqrt 3$$
Exercise 6: Master the Concepts
Important clarifications concerning polynomials:
A
If an $n$th degree polynomial and an $n$th degree equation have equal coefficients, what is then actually the difference between them?
How many real roots does a complex polynomial of degree $n$ have?
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?
For two $n$th degree polynomials $P$ and $Q\,$ it is known that every root of $P$ is a root of $Q$ with equal algebraic multiplicity. Are the two polynomials identical?
answer
Second bullet point: Between $0$ and $n$.
Third bullet point: If even degree, between $0$ and $n$. If odd degree, between $1$ and $n$. (Both included.)
Fourth bullet point: $n$.
Fifth bullet point: No. Example: $Q(z)=kP(z), k \neq 0,1$.
Determine the derivative of the function $\,P(x)\,.$
answer
$$P'(x)=100x^3-99x^2+98x-97$$
B
In section 1.10 of eNote 1 about complex numbers, differentiation of complex functions of a real variable is introduced. An example of a complex function of a real variable is a function $\,f\,$ such as the following:
Note that the real part and the imaginary part of $\,f\,$ each are a real polynomial function of a real variable. Determine the derivative $\,f’(t)\,$, and compute the value $\,f’(0)\,.$
hint
See Theorem 1.60 in eNote 1 along with the subsequent examples.
Show that $\,Q\,$ can be differentiated following the same rules that apply to real polynomials of a real variable, if just $\,i\,$ is treated as any other constant.
hint
Differentiate real part and imaginary part individually. Then start over and differentiate the function without splitting into rectangular form. Do both methods arrive at the same result?
Exercise 8: Polynomial with Complex Coefficients (Advanced)
A
Solve the binomial quadratic equation $\,z^2=3-4i\,.$
hint
Use the method used in Example 2.23 in eNote 2.
answer
$$-2+i\quad\text{ and }\quad 2-i$$
B
Given the polynomial
$$\,P(z)=z^2-(1+2i)z-\frac 32+2i\,.$$
Compute the roots of the polynomial.
answer
$$\frac 32 +\frac 12\,i\quad\text{ and }\quad-\frac 12 +\frac 32\,i$$
Exercise 9: The Geometry of a Quadratic Equation. Enjoy!
We are given the complex number $\,\alpha = 3-4i\, $ and the complex set of numbers
