Partial Fraction

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

Question 4:

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

Solution:

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

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

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

put x = 1

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

 1 = A(3)² + B(0) + C (0)

 1 = 9 A

  A = 1/9

put x = -2

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

1 = [A(0)²] + [B(-3)(0)] + [C(-3)]

1 = 0 + 0 - 3 C

-3 C = 1

    C = -1/3

put x = 0

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

1 = [A(2)²] + [B(-1)(2)] + [C(-1)]

1 = 4A - 2 B - C

1 = 4(1/9) - 2 B - (-1/3)

1 = (4/9) - 2 B + (1/3)

1 = (4 - 18 B + 3)/9

1 = (7 - 18 B)/9

9 = 7 - 18 B

 9 - 7 = - 18 B

   - 2 = - 18 B

     B = 2/18

     B = 1/9

  = [1/9(x - 1)] + [1/9(x + 2)] -[1/3(x + 2)²]


Question 5:

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

Solution:

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

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

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

put x = 1

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

 -1 = A(0)² + B(0) + C (3)

 -1 = 3 C

  C = -1/3

put x = -2

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

-4 = [A(-3)²] + [B(-3)(0)] + [C(0)]

-4 = 9 A + 0 + 0

 9 A= -4

    A = -4/9

put x = 0

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

-2 = [A(-1)²] + [B(-1)(2)] + [C(2)]

-2 = A - 2 B + 2 C

-2 = (-4/9) - 2 B + 2(-1/3)

-2 = (-4/9) - 2 B - (2/3)

-2 + (4/9) + (2/3) = - 2 B

(-18 + 4 + 6)/9 = - 2 B

 - 8/9 = -2 B

    B = 8/(9 x 2)

    B = 4/9

= [-4/9(x + 2)] + [4/9(x - 1)] + [-1/3(x - 1)²]