Partial Fractions  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 factored 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 Partial Factions Example Consider the rational function                       3x + 2                   3x +2         P(x) =                    =                                                        x2 - 1               (x - 1)(x + 1) We want to write it in the form                  3x + 2                       A                 B                                        =                  +                             (x - 1)(x + 1)              x - 1            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 + 2               5/2              1/2                             =                   +                             x2 - 1              x - 1             x + 1 This is called the partial fraction decomposition of P(x) Example 2 Find the Partial Fraction Decomposition of                        3x2 + 4x + 7               3x2 + 4x + 7         P(x) =                              =                                                             x3 - 2x2 + x                    x(x-1)2                                                  We write                3x2 + 4x + 7               A                    B                   C                                       =                     +                       +                                x(x-1)2                  x - 1             (x - 1)2              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              - 4                  14                   7                                       =                     +                       +                                x(x-1)2                  x - 1             (x - 1)2              x Integration Example:  Evaluate         We write                x2 - 2                    A              Bx + C                                                =                 +                                       x(x2 +1)                 x                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         Exercise Find         Logistics Growth It has been said that the total population of South Lake Tahoe should never exceed 30,000 people.  If in 1980 the population was 15,000 and in 2000 it was 20,000, when will the population reach 25,000? Solution:   We make the assumption that the rate of increase of the population is proportional to the product of the current population and 30,000 minus the current population. That is:            dP                     =  kP(30,000 - P)                dt  This is a separable differential equation.  Separating gives                     dP                                          = kdt           P(30,000 - P) Now integrate both sides.  The right hand side is          kt + C To integrate the right hand side, use partial fractions:                  1                       A                   B                                       =             +                                      P(30,000 - P)            P              30,000 - P          Multiplying by the common denominator:         1 = A(30,000 - P) + BP           P = 0:      1 = 30,000A     or     A = 1/30,000         P = 30,000:  1 = 30,000B     or     B = 1/30,000 Now integrate to get              1                      1                        lnP -                ln(30,000 - P)  =  kt + C          30,000              30,000 or                      P              ln                          =  at + b               30,000 - P When t = 0, P = 15,000, so         b = 0 When t = 20, P = 20,000, so         20a = ln2  or                      ln 2          a =                                        20 hence                      P                        t ln 2         ln                          =                                     30,000 - P                  20 Finally                     t ln 2         ln 5 =                                         20 or                 20 ln 5         t =                    = 46.4                    ln 2 The population will reach 2500 in the year 2026. Partial Fractions Exercises ExercisesA.  B .  C.  D.    e-mail Questions and Suggestions