Partial Fraction Practice Questions

In this page partial fraction practice questions we are going to see some practice questions with solution.

Question 1:

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

Solution:

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

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

                 1 = [A(x + 1) + B(x - 1)]

put x = 1

 1 = [A(1 + 1) + B(1 - 1)]

 1 = [A(2) + B(0)]

2 A = 1

  A = 1/2

put x = -1

 1 = [A(-1 + 1) + B(-1 - 1)]

 1 = [A(0) + B(-2)]

-2 B = 1

  B = -1/2

 1/[(x -1) (x + 1)] = [1/2(x - 1)] - [1/2(x + 1)]   


Question 2:

Resolve into partial fractions (7 x - 1)/[6 - 5 x + x²]

Solution:

(7 x - 1)/[6 - 5 x + x²]

First we have to factorize the quadratic equation which is in the denominator

6 - 5 x + x² = (x - 2) (x - 3)

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

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

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

put x = 3

 7 (3) - 1 = [A(3 - 3) + B(3 - 2)]

 21 - 1 = [A(0) + B(1)]

  20 = B

  B = 20

put x = 2

 7 (2) - 1 = [A(2 - 3) + B(2 - 2)]

 14 - 1 = [A(-1) + B(0)]

 13 = -A

  A = -13

(7 x - 1)/[6 - 5 x + x²]  = [-13/(x - 3)] + [20/(x - 2)]   


Question 3:

Resolve into partial fractions  (x² +  x + 1)/[(x - 1) (x - 2) (x - 3)]

Solution:

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

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

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

put x = 1

(1² + 1 + 1)=[A(1 - 2)(1 - 3)]+[B(1 - 1)(1 - 3)]+[C(1 - 1)(1 - 2)]

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

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

3 = 2 A + 0 + 0

2 A = 3

 A = 3/2

put x = 2

(2² + 2 + 1)=[A(2 - 2)(2 - 3)]+[B(2 - 1)(2 - 3)]+[C(2 - 1)(2 - 2)]

(4 + 2 + 1)=[A (0) (-1)] + [B (1) (-1)] + [C (1) (0)]

7 = 0 - B + 0

- B = 7

  B = -7

put x = 3

(3² + 3 + 1)=[A(3 - 2)(3 - 3)]+[B(3 - 1)(3 - 3)]+[C(3 - 1)(3 - 2)]

(9 + 3 + 1)=[A (1) (0)] + [B (2) (0)] + [C (2) (1)]

13 = 0 + 0 + 2 C

13 = 2 C

C = 13/2

= [3/2(x - 2)(x - 3)]-[7/(x - 1)(x - 3)]+[13/2(x - 1)(x - 2)]

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