PARTIAL FRACTION DECOMPOSITION WITH REPEATED FACTORS
Express the following as a sum of partial fractions.
Problem 1 :
Solution:
5x2 + 17x + 15 = A(x + 2)(x + 1) + B(x + 1) + C(x + 2)2
5x2 + 17x + 15 = A(x2 + x + 2x + 2) + Bx + B + C(x2 + 4x + 4)
= Ax2 + Ax + 2Ax + 2A + Bx + B + x2C + 4xC + 4C
Equating the coefficients of x2, x and constants.
5x2 + 17x + 15 = x2(A + C) + x(3A + B + 4C) + (2A + B + 4C)
A + C = 5 ---> (1)
3A + B + 4C = 17 ---> (2)
2A + B + 4C = 15 ---> (3)
Subtracting (3) from (2)
A = 2
By applying the value of A in the (1) equation,
2 + C = 5
C = 3
By applying the value of A and C in the (2) equation,
3(2) + B + 4(3) = 17
6 + B + 12 = 17
18 + B = 17
B = -1
Problem 2 :
Solution:
Problem 3 :
Solution:
x2 + 1 = A(x - 1)2 + B(x - 1) (x + 1) + C(x + 1)
= A(x2 + 1 - 2x) + B(x2 - 1) + C(x + 1)
= x2A + A - 2xA + x2B - B + Cx + C
x2 + 1 = x2(A + B) + x(-2A + C) + (A - B + c)
Equating the coefficients of x2, x and constants.
A + B = 1 ---> (1)
-2A + C = 0 ---> (2)
A - B + C = 1 ---> (3)
Subtracting (2) and (3),
-3A + B = -1 ---> (4)
Subtracting (1) and (4),
A = 1/2
By applying the value A in (1),
1/2 + B = 1
B = 1 - 1/2
B = 1/2
By applying the values A and B in (3),
1/2 - 1/2 + C = 1
C = 1
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