PARTIAL FRACTIONS DECOMPOSITION EXAMPLES

Example 1 :

Resolve the following rational expressions into partial fractions.

1/(x2-a2)

Solution :

Let us decompose the denominator into linear factors.

1/(x2-a2)  =  [A(x + a) + B(x - a)]/(x2-a2)

 =  A(x + a) + B(x - a)

When x = -a

1  =  B(-a –a)

1  =  B(-2a)

B = -1/2a

When x = a

1  =  A(a +a)

1  =  A(2a)

A = 1/2a

Hence the solution is

Example 2 :

Resolve the following rational expressions into partial fractions.

(3x + 1)/(x - 2) (x  + 1)

Solution :

3x + 1  =  A(x + 1) + B(x - 2)

When x = -1

3(-1) + 1  =  B(-1 - 2)

-3 + 1  =  B(-3)

-2 =  -3B

B  =  2/3

When x = 2

3(2) + 1  =  A(2 + 1)

6 + 1  =  A(3)

7 =  3A

A  =  7/3

Hence the solution is

Example 3 :

Resolve the following rational expressions into partial fractions.

x/(x2 + 1)(x - 1)(x + 2)

Solution :

x  = A(x+2)(x2+1) + B(x2+1)(x-1) + (Cx + D)(x-1)(x+2)

When x = 1

1   =  A(3)(2)

1  =  6A

A  =  1/6

When x = -2

-2   =  B(5)(-3)

-2  =  -15B

B  =  2/15

When x = 0

x  = A(x+2)(x2+1) + B(x2+1)(x-1) + (Cx + D)(x-1)(x+2)

0  =  A(2)(1) + B(1)(-1) + D(-1)(2)

0  =  2A - B - 2D

By applying the values of A and B, we get

0  =  (1/3) - (2/15) - 2D

2D  =  3/15

D  =  1/10

When x = -1

-1  = A(1)(2) + B(2)(-2) + (-C+D)(-2)(1)

-1  = 2A - 4B + 2C - 2D

By applying the values of A, B and D

-1  = (1/3) - (8/15) + 2C - (1/5)

-1  =  ((5 - 8 - 3)/15) + 2C

-1  =  -6/15 + 2C

-1 + (2/5)  =  2 C  ==> -3/5  =  2C  ==> C  =  -3/10

Hence the solution is

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. SAT Math Videos

    May 22, 24 06:32 AM

    sattriangle1.png
    SAT Math Videos (Part 1 - No Calculator)

    Read More

  2. Simplifying Algebraic Expressions with Fractional Coefficients

    May 17, 24 08:12 AM

    Simplifying Algebraic Expressions with Fractional Coefficients

    Read More

  3. The Mean Value Theorem Worksheet

    May 14, 24 08:53 AM

    tutoring.png
    The Mean Value Theorem Worksheet

    Read More