Partial Fraction Decomposition: Summary and Higher Powers

in mathematics •  last month  (edited)

In this video I summarize the basic linear and quadratic factors cases for partial fraction decomposition as well as talk briefly about factors with higher powers. It is also interesting to note that the 20 year old Evariste Galois proved that for factors with powers of 5 or higher there is no general formula for partial fraction decomposition nor will there ever exist any. Also of note is the Evariste died in a duel when he was only 20 years old.

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Partial Fraction Decomposition: Summary and Higher Powers

Partial Fractions Summary.jpeg

In my earlier videos I went over the proof for the partial fraction decomposition of the simple linear cases.

For all the following theorems, assume the following:

  • N and D are polynomials where degree of N is smaller than degree of D
  • K is a constant
  • a1, … , aj are all different
  • m1, … , mj , n1, … nk are all positive integers
  • x2 + b1x + c1, … , x2 + bkx + ck are all different.

Simplest Case (proved in my earlier video)

  • Linear factors that don't repeat

General Case for Linear Factors (proved in my earlier video)

  • Linear factors that repeat

Simple Linear and Quadratic Factor Case

  • Combination of non-repeating linear factors with non-repeating irreducible factors of degree 2
  • The proof for this is similar to the proofs in my earlier videos but much more tedious so I will not cover them anytime soon (maybe sometime in the future)

General Linear and Quadratic Factor Cases

  • Combination of repeating linear factors with repeating irreducible factors of degree 2

Note on Factors of Higher Polynomials

There are similar theorems for factors of higher power polynomials but for degree 5 or higher, the 20 year old French mathematician Evariste Galois proved that simple general formulas do not exist and cannot ever be found.

Interesting Notes on Evariste Galois

  • Lived from October 25, 1811 to May 31, 1832.
  • In his teens, he was able form several important proofs and theorems in advanced mathematics and algebra.
  • He was a political prisoner during the political turmoil in France during his time.
  • He died during a duel.
    • The reasons for the duel have not been fully discovered with many possible theories presented with some pointing to letters he wrote shortly prior to his death about a broken love-affair.
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