Resolve the following rational expressions into partial fraction. (x-1)²/ x³+x.
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Answer:
The expression can be written as:
(
x
−
1
)
2
x
3
+
x
=
x
2
−
2
x
+
1
x
(
1
+
x
2
)
This gives us a linear and a quadratic factor in the expression.
x
2
−
2
x
+
1
x
(
x
2
+
1
)
=
a
0
x
+
a
2
x
+
a
1
x
2
+
1
Now, we multiply the entire equation by the denominators of left-hand side to get:
x
2
−
2
x
+
1
=
a
0
(
x
2
+
1
)
+
x
(
a
2
x
+
a
1
)
We now have to value of the parameters. Substituting
x
=
0
gives:
0
2
−
2
⋅
0
+
1
=
a
0
(
0
2
+
1
)
+
0
(
a
2
⋅
0
+
a
1
)
a
0
=
1
We now substitute this value of the parameter in the equation, expand the equation, and then regroup it to compare the coefficients on either side of the equation.
x
2
−
2
x
+
1
=
1
(
x
2
+
1
)
+
x
(
a
2
x
+
a
1
)
x
2
−
2
x
+
1
=
x
2
+
1
+
a
2
x
2
+
a
1
x
x
2
−
2
x
+
1
=
x
2
(
a
2
+
1
)
+
a
1
x
+
1
Comparing the coefficients, we get:
a
1
=
−
2
1
+
a
2
=
1
a
2
=
0
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