## Sample Reflection Essay (for an extra Calculus Token)

### Title: Partial Fraction Decomposition Lecture

A majority of the first half of the lecture involved concepts/techniques taught and used in differential equations and linear algebra, so I had trouble understanding them. For example, matrixes were used in order to divide 2 polynomials together (even though they did not necessarily share a root). However, I did understand the main idea, which was trying to reduce polynomials to irreducible parts, so that they could be divided and simplified. Even though the first half was harder to understand, the main idea held true throughout the second half as well. In reference to this first half, which included the formal definition of partial fraction decomposition with relations to polynomials, I asked the professor which courses he thought the theorem should be taught in. He responded that it does not have a place in the vast majority of courses because it is algebra, yet requires a lot of analysis and higher level concepts. It is currently taught in numbers theory and may have a place in abstract algebra.

In the second half of the lecture, the Professor showed a much more simplified and practical use of partial fraction decomposition. For example, 1/15 can be rewritten as (a/3) + (b/5). When A and B are solved for, you get (2/3) - (3/5). Now, how does this help? Well, theoretically, ancient Greeks could have used this to make a 15-gon shape if they knew how to make an equilateral triangle and a regular pentagon inscribed in a circle. You can travel 2/3 of the circle forward (2 sides of the triangle) and then back 3/5 (3 sides of the pentagon), leaving you with a side that is exactly 1/15 of the circle. Using this technique, various intricate shapes can be created. This technique also has many uses in algebra and calculus. Personally, I found the second half of the lecture to be very interesting because the technique seemed so simple and useful, yet I had never thought of it before.