resolve into denominator x/(x+3)(x-4).please answer my question
Answers
Answer:
First we will factor the denominator as much as possible:
x
4
+
1
x
3
(
x
2
+
4
)
And now, we will choose the factors to write:
x
4
+
1
x
3
(
x
2
+
4
)
=
A
x
+
B
x
2
+
C
x
3
+
D
x
+
E
x
2
+
4
Note that since there was a lone power of
x
, (which was
x
3
) I wrote out successive powers of
x
, starting at
x
to the first, and ending at
x
3
. There was also a quadratic term,
x
2
+
4
, which couldn't be factored - so for that one, we used
D
x
+
E
in the numerator.
The next step is to multiply both sides of the equation by
x
3
⋅
(
x
2
+
4
)
and cancel off what we can:
x
4
+
1
=
A
x
2
⋅
(
x
2
+
4
)
+
B
x
⋅
(
x
2
+
4
)
+
C
⋅
(
x
2
+
4
)
+
x
3
(
D
x
+
E
)
And now, we will distribute and simplify everything:
x
4
+
1
=
A
x
4
+
4
A
x
2
+
B
x
3
+
4
B
x
+
C
x
2
+
4
C
+
D
x
4
+
E
x
3
We can solve for each constant now, by using the technique of grouping. The first step is to rearrange everything in successive powers of
x
:
x
4
+
1
=
A
x
4
+
D
x
4
+
B
x
3
+
E
x
3
+
4
A
x
2
+
C
x
2
+
4
B
x
+
4
C
And now, we will factor out the constant terms:
x
4
+
1
=
(
A
+
D
)
x
4
+
(
B
+
E
)
x
3
+
(
4
A
+
C
)
x
2
+
4
B
x
+
4
C
The next step is to create a system of equations using the coefficients of
x
on the left side that correspond to the coefficients of
x
on the right side. What do I mean? Well, we can see that there is a term
1
⋅
x
4
on the left side, but there is also a
(
A
+
D
)
⋅
x
4
term on the right side.
This implies that
A
+
D
=
1
. We will continue in this manner, building a system using all the coefficients:
A
+
D
=
1
B
+
E
=
0
4
A
+
C
=
0
4
B
=
0
4
C
=
1
Immediately from the last two equations, we can conclude that
B
=
0
and
C
=
1
4
.
From this it follows that since
B
+
E
=
0
,
E
must also equal
0
. And since
4
A
+
C
=
0
,
A
must equal
−
1
16
.
Then, after plugging
A
into the last equation
A
+
D
=
1
, and solving for
D
, we obtain
D
=
17
16
.
Now all that's left is to plug these coefficient values into our expanded expression:
x
4
+
1
x
3
(
x
2
+
4
)
=
A
x
+
B
x
2
+
C
x
3
+
D
x
+
E
x
2
+
4
x
4
+
1
x
3
(
x
2
+
4
)
=
1
4
x
3
+
17
x
16
(
x
2
+
4
)
−
1
16
x
And there we have it. Remember, successfully expanding with partial fractions is all about choosing the correct factors, and from there it's just a lot of algebra. If you are familiar with the grouping technique, then you shouldn't have any trouble solving for the coefficients.