Partial Fraction Decomposition: Proof of the Simplest Case: REVISITED

In this video I revisit my earlier video on “Partial Fraction Decomposition: Proof of the Simplest Case”, https://youtu.be/Eq5yLJmC6pw, because a person by the username Heliocentric brought to my attention that I had not fully completed the proof. After a few back and forth exchanges I quickly realized that indeed Heliocentric was right and I had still some part of the proof yet to be explained. The issue at hand is in the Lemma 1 part of the proof which required proving that a polynomial exists with some given characteristics. In that video though, I simply assumed that the polynomial existed and worked to prove the characteristics. This is not fully complete because I still had yet to prove that it could have been in fact a polynomial. But as Heliocentric stated, and as I promised him 6 months ago that I would complete the proof of this video “later” (i.e. finally today!), I show how the proof that the function is in fact a polynomial can be shown by the Factor Theorem. Now recall in my earlier videos that the Factor Theorem is just a special case of the Polynomial Remainder Theorem, in which the division of an univariant polynomial by (x – a) has a remainder that equals zero. This in fact is the case with the formulation in the Lemma 1, and hence the factor theorem proves that in fact there does exist a polynomial P(x) with the given characteristics. This is a very important video illustrating now only the importance of thoroughly proving each step of a derivation, but also the importance to learn from constructive feedback and always make sure to correct yourself, so make sure to watch this video!

Video notes and playlist:

- PDF notes: https://1drv.ms/b/s!As32ynv0LoaIh44BB2VefNaDHI-HDQ
- HIVE notes: https://peakd.com/mathematics/@mes/partial-fraction-decomposition-proof-of-the-simplest-case-revisited
- Partial Fractions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HrJh8AbOkh3oYxSnthP997

Related Videos:

Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw
Polynomial Remainder Theorem: Factor Theorem & Factorization: https://youtu.be/ThX3uSXCD14
Euclidean Division of Integers: Theorem and Proof: https://youtu.be/66juubotzi0
Euclidean Division of Polynomials: Theorem and Proof: https://youtu.be/ONxn17okl5c
Polynomial Remainder Theorem: Proof + Factor Theorem: https://youtu.be/q4lwSBObkXc
Polynomial Remainder Theorem: Elementary Proof: https://youtu.be/Im0c25WbZBg .

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