Problem 1
a) Find the (complex) roots of the polynomial p(x) = 2x2 + 12x + 26.
b) Find the values of parameter b for which the equation bx2 − bx + 2 = 0 has no real roots.
c) Find all roots (real or complex) of the polynomial p(x) = x6−x5−3x4−3x3−22x2+4x+24
.
Hint: x = 3 is a root. Divide out the associated linear factor and continue with more roots that are easy to guess.
Problem 2
Assuming that z = a + ib is a complex number, compute real and imaginary parts of a)
b)
c) (z∗)2z
Bonus: |x| is the absolute value function:√
In the case of x ∈ C: |x| = √xx∗
In the case of x ∈ R: |x| = x2 or in other words |x| = x if x ≥ 0; |x| = −x if x < 0. In both cases |x| ∈ R and |x| ≥ 0.
d) Compute . Use the definition of the absolute value function for complex numbers.
d) Characterize the set of real numbers x that satisfy |4x + 2| ≤ |2x − 3| .
Hint: You cannot directly work with | |. Use the definition of absolute value for real numbers to change the inequality into an equivalent problem without | |. For that, you can apply certain functions to both sides of the inequality without changing the inequality.
Problem 3
(4+3+3 points)
Proof the following for complex numbers z and w, i.e. z,w ∈ C. a)
b) Re(
c) Im(
Re(z) and Im(z) are the real and complex part of z, respectively. I.e. if z = a + bi, then Re(z) = a and Im(z) =
