Now
that you've gone through our explanation on the idea of partial fractions, take
this short quiz to find our how much you have learnt.
Express the following in partial
fractions:
Let
By cover-up
rule,
D = 1 / (1 + 1)
= 0.5
(x - 1)2(Ax
+ B) + C (1 + x2) (x - 1) + 0.5 (x2
+ 1) = 1
x3
(A + C) + x2 (0.5 - C - 2A + B) + x
(A + C - 2B) + B - C + 0.5 = 1
Comparing
coefficient of x3 term:
A + C = 0
A = -C
Comparing
coefficient of constant term:
B - C + 0.5 = 1
B = C + 0.5
Comparing
coefficient of x term:
A + C - 2B = 0
-C + C - 2 (C + 0.5) =
0 substituting
above equations
C = -0.5
A = 0.5, B = 0
Hence,
Let
From
above identity, A = 2
Using cover-up rule,
C = (2+2-3+8) / (1+2)
= 3
2x3
+ 2x2 - 3x + 8 = (2x + B)(x - 1)(x
+ 2) + 3(x + 2) - 2(x - 1)
Let x
= 0,
-2B + 6 + 2 = 8
B = 0
Hence,
Let
From
identity, A = 1
Using
"cover-up" rule,
2x4
+ 5x3 +12x2 + 20x - 1 = (x + B)(x2
+ 4)(2x - 1) + (Cx + D)(2x -1) + 3(x2 +
4)
= 2x4 + x3(2B - 1) + x2(2C
- B + 11) + x(8B - C + 2D -4)
+ 12 - 4B - D
Compare coefficient of
x3:
2B - 1 = 5
B = 3
Compare constant term:
12 - 4B - D = -1
D = 1 substituting
B = 3
Compare coefficient of
x2:
2C - B + 11 = 12
C = 2 substituting
B = 3
Hence,
