5. The partial fractions of
x 3 - 5
x² – 3x+2
are
4
3
2x+3+
x-1
x-2
4 3
1)x+ 3 -
+
x-1 x-2
4 2.
3) x + 3 -
x-1 x-2
4 3
4) x + 3+ +
x-1 x-2
Answers
Answered by
3
As the degree of the numerator is
>
than the degree of the denominator, we do a long division
a
a
a
a
2
x
3
−
x
2
+
x
+
5
a
a
a
a
∣
x
2
+
3
x
+
2
a
a
a
a
2
x
3
+
6
x
2
+
4
x
a
a
a
a
a
a
∣
2
x
−
7
a
a
a
a
a
0
−
7
x
2
−
3
x
+
5
a
a
a
a
a
a
a
−
7
x
2
−
21
x
−
14
a
a
a
a
a
a
a
a
a
a
a
a
a
a
18
x
+
19
Therefore,
2
x
3
−
x
2
+
x
+
5
x
2
+
3
x
+
2
=
2
x
−
7
+
18
x
+
19
x
2
+
3
x
+
2
We can now do the decomposition into partial fractions
18
x
+
19
x
2
+
3
x
+
2
=
18
x
+
19
(
x
+
1
)
(
x
+
2
)
=
A
x
+
1
+
B
x
+
2
=
A
(
x
+
2
)
+
B
(
x
+
1
)
(
x
+
1
)
(
x
+
2
)
So,
18
x
+
19
=
A
(
x
+
2
)
+
B
(
x
+
1
)
Let
x
=
−
1
,
⇒
,
1
=
A
Let
x
=
−
2
,
⇒
,
-17=-B#
2
x
3
−
x
2
+
x
+
5
x
2
+
3
x
+
2
=
2
x
−
7
+
1
x
+
1
+
17
x
+
2
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