[Solved] Calculus-Homework 1

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Problem 1

  1. Find the (complex) roots of the polynomial p(x) = 2x2 + 12x + 26.
  2. Find the values of parameter b for which the equation bx2 bx + 2 = 0 has no real roots.
  3. Find all roots (real or complex) of the polynomial p(x) = x6x5−3x4−3x3−22x2+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)


  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.


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[Solved] Calculus-Homework 1
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