Problem 1

26. Find the (complex) roots of the polynomial p(x) = 2x² + 12x + 26.
27. Find the values of parameter b for which the equation bx² − bx + 2 = 0 has no real roots.
28. 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)

1. (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.

1. Compute . Use the definition of the absolute value function for complex numbers.
2. 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

Proof the following for complex numbers z and w, i.e. z,w ∈ C. a)

1. Re(
2. 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) = b.