Partial Fractions Part I

1. The Fundamental Theorem of Algebra
   The fundamental theorem of algebra states that if P(x) is a polynomial of degree n then P(x) can be factored into linear factors over the complex numbers.  Furthermore, P(x) can be factor over the real numbers as a product of linear and quadratic terms and any rational function can be split up as a sum of rational function with denominators of the form (x - r)ⁿ or x² + Ax + B. 
   

2. Partial Factions

   Example:

   Consider the rational function

   P(x) = (3x + 2)/(x² -1) = 3x/[(x - 1)(x + 1)]

   We want to write it in the form

   (3x + 2)/[(x - 1)(x + 1)] = A/(x - 1) + B/(x + 1)

   To do this we need to solve for A and B.  Multiplying by the common denominator

   (x - 1)(x + 1) we have

   3x  + 2 = A(x + 1) + B(x - 1)  

   Now let x = 1

   5 = 2A + 0

   A = 5/2

   Now let x = -1

   -1 = -2B

   B = 1/2

   Hence we can write

   3x/(x² -1) = (5/2)/(x - 1) + (1/2)/(x + 1)

   This is called the partial fraction decomposition of P(x)

   Example 2:

   Find the Partial Fraction Decomposition of

   P(x) = (3x² + 4x + 7)/(x³ - 2x² + x) =  (3x² + 4x + 7)/[x(x-1)²]

   We write

   (3x² + 4x + 7)/[x(x-1)²] = A/(x - 1) + B/(x - 1)² + C/x

   Multiplying by the common denominator, we have

   A(x(x - 1)) + Bx + C(x - 1)²  = 3x² + 4x + 7

   Let x = 0:

   C = 7

   Let x = 1:

   We have B = 14

   Now look at the highest degree coefficient:

   Ax² + Cx²  = 3x²

   Dividing by x² and substituting C = 7

   A + 7 = 3, A = -4

   We conclude that

   (3x² + 4x + 7)/[x(x-1)²] = -4/(x - 1) + 14/(x - 1)² + 7/x
   

3. Integration

   Example:  Evaluate

   int (x² -2)/[x(x² + 1)]dx

   We write

   (x² -2)/[x(x² + 1)] = A/x + (Bx + C)/(x² + 1)

   Multiplying by the common denominator, we have

   A(x² + 1) + (Bx + C)x = x² - 2

   Let x = 0

   A = -2

   Hence

   (Bx + C)x = x² - 2 + 2x² + 2 = 3x²

   So that

   Bx² + Cx = 3x²

   We see that B = 3 and C = 0

   Hence

   int (x² -2)/[x(x² + 1)]dx = int -2/x + (3x )/(x² + 1)dx

   = -2int 1/x dx + 3 int x/(x² + 1)dx u = x² + 1 du = 2xdx

   = -2ln|x| + 3/2ln(x² + 1) + C

   Exercise:

   Find

   int 1/(x⁴ - 1) dx