Partial Fractions

Partial Factions

In algebra we learn how to find the common denominator of two expressions and then combine like terms.  Partial fractions are all about going backwards.

Example 1

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 + 24x + 7                 3x² + 24x + 7
        P(x)  =                                 =                                  
                        x³ - 2x² + x                        x (x - 1)²

Since we have a square, we need to write the decomposition in the form below.

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

Multiplying by the common denominator, we have

        Ax(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 the integral

Solution

First we find the partial fraction decomposition of the integrand.  We factor to get

                4x + 2                      A                B
                                     =                  +               
           (x - 2)(x + 3)               x - 2           x + 3

Multiply by the common denominator to get

        A(x + 3) + B(x - 2)  =  4x + 2

Let  x  =  2:

        5A  =  10        A  =  2

Let  x  =  -3:

        -5B  =  -10        B  =  2

Now we can integrate

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                    
                                        =   k dt
            P(30,000 - P)

Now integrate both sides.  The integral of the right hand side is 

        kt + C

To integrate the left hand side, use partial fractions:

                       1                      A               B
                                        =           +                     
             P(30,000 - P)              P          30000 - P


Multiplying by the common denominator:

        A(30,000 - P)  +  BP  =  1

        P = 0:      1  =  30,000A     or     A  =  1/30,000

        P = 30,000:  1  =  30,000B     or     B  =  1/30,000

Now integrate, noticing that the denominator 30,000 - P will result in an added negative sign after substitution.  We get

               1                     1
                        lnP -                  ln(30,000 - P)  =  kt + C 
          30,000              30,000

or

        =  at + b    a  =  30000k      b  =  30000C

When t  =  0,   P  =  15000, so

        b  =  0

When t = 20, P = 20,000, so

        20a  =  ln2 

or 

                    ln 2
        a  =                
                     20

hence

       

Finally plugging in 25,000 for P gives

                      t ln 2
        ln 5  =                 
                        20

or

                 20 ln 5
        t  =                   =  46.4 
                    ln 2

The population will reach 25,000 in the year 2026.

Exercise

On Monday the Ebola virus is caught by 3 people in an isolated community of 10,000 people.  By Thursday, 40 people have the virus.  How long will it be until half of the population have the virus?

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