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Week 3, Long Day: Polynomials

Polynomials are central to complex numbers. It all started with polynomials. Why do certain polynomials have roots, and which rules apply to the question of how many roots different polynomials have? These are questions that occupied the mathematicians that developed the theory of complex numbers. They found that if you introduce a new number $i\,$ that satisfies $\,i^2=-1\,,$ then every polynomial has at least one root. And more spectacularly: If you count the roots with multiplicity then every $n$’th degree polynomial has exactly $n$ roots. Today we will study the structure of polynomials and their roots, including strategies for finding the roots. We will also look at complex-valued functions (of a real variable) whose real and imaginary parts are real polynomials. Can these functions be differentiated?

Today’s Key Concepts

Polynomials versus algebraic equations. Roots of linear polynomials. Solution methods for binomial and general quadratic equations with real and complex coefficients, respectively. Roots of polynomials. Theorem of descent. Factorization of polynomials (the rule of zero product is decisive!). Algebraic multiplicity. Differentiation of polynomials.

Preparation and Syllabus

Today’s topics are from eNote 2 Polynomials. But we will be returning to subjects from eNote 1 Complex Numbers, in particular section 1.10, to become able to differentiate complex functions.

Activity Program

  • 10.00 – 12.00: Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.30 – 17.00: Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: Your teachers are present in the study areas

Remarks about Homework Assignments

Homework set 1 (HW1) is available (find it via a link on the page “Autumn2022”). The deadline for hand-in of your solution to HW1 is Sunday September 25th, 23:55. We would advise you to begin working on the assignment well in advance - note in particular that we have the first Theme Exercise 1 in the same week (the Friday of next), so plan ahead.

Your teachers will give a briefing today in the study areas in the afternoon about homework and theme exercises so you are prepared and know how and where to hand in as well as what to be aware of.

Group Exercises

  1. Today’s Wetware Exercise
  2. Linear Polynomials
  3. Binomial Quadratic Equations with Real Right-Hand-Sides
  4. Factorizations
  5. The Theorem of Descent
  6. Master the Concepts
  7. Differentiation
  8. Polynomials with Complex Coefficients (Advanced)
  9. The Geometry of a Quadratic Equation. Enjoy!