Partial Fractions Example Problems

In this page partial fractions example problems we are going to see some example problems based on the topic partial fractions.

Question 6:

Resolve into partial fractions (x + 1)/[(x - 2)² (x + 3)]

Solution:

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

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

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

 x = 2

2 + 1 = A(2 - 2)(2 + 3) + B(2 + 3) + C(2 - 2)²

3 = A(0)(5) + B(5) + C(0)

 3 = 0 + 5 B + 0

 5 B = 3

   B = 3/5

put x = -3

-3 + 1 = A(-3 - 2)(-3 + 3) + B(-3 + 3) + C(-3 - 2)²

 -2 = [A(-5) (0)] + [B(0)] + [C(-5)²]

-2 = A(0) + B(0) + C(25)

-2 = 0 + 0 + 25 C

 25 C = - 2

    C = -2/25

put x = 0

0 + 1 = A(0 - 2)(0 + 3) + B(0 + 3) + C(0 - 2)²

1 = A(-2)(3) + B (3) + C(-2)²

1 = -6A + 3B + 4C

1 = -6A + 3(3/5) + 4(-2/25)

1 = -6A + (9/5) - (8/25)

1 = (-150A+45-8)/25

25 = -150 A + 37

25 - 37 = -150 A

- 12 = -150 A

  A = 12/150

  A = 2/25

         = [2/25(x - 2)] + [3/5(x - 2)²] - [2/25(x + 3)]


Question 7:

Resolve into partial fractions (x² - 6 x + 2)/[x² (x + 2)]

Solution:

(x² - 6 x + 2)/[x² (x + 2)] = [A/x] + [B/x²] + [C/(x + 2)]

(x² - 6 x + 2)/[x² (x + 2)] = [Ax(x + 2) + B(x + 2) + Cx²]/[x² (x + 2)]

(x² - 6 x + 2) = [Ax(x + 2) + B(x + 2) + Cx²]

 x = 0

(0² - 6 (0) + 2) = [A(0)(0 + 2) + B(0 + 2) + C(0)²]

2 = 0 + 2 B + 0

2 B = 2

  B = 1

put x = -2

((-2)² - 6 (-2) + 2) = [A(-2)(-2 + 2) + B(-2 + 2) + C(-2)²]

(4 + 12 + 2) = A(-2)(0) + B(0) + C(4)

18 = 0 + 0 + 4 C

18 = 4 C

 C = 4/18

 C = 2/9

put x = 1

(1² - 6(1) + 2) = [A(1)(1 + 2) + B(1 + 2) + C(1)²]

1 - 6 + 2  = A(1)(3) + B (3) + C(1)

3 - 6 = 3 A + 3B + C

- 3 = 3 A + 3 (1) + 2/9

- 3 = 3 A + 3 + 2/9

 - 3 = (27 A + 27 + 2)/2

- 6 = 27 A + 29

-6 - 29 = 27 A

 -35 = 27 A

  A