Answer:
Step-by-step explanation:
Solution set is
{
−
4
,
3
+
i
2
,
3
−
i
2
}
Step-by-step explanation:
NOTE: I WROTE THIS ANSWER THINKING THE QUESTION WAS TO SOLVE FOR
x
, NOT JUST TO FIND THE ROOTS
Step 1: List the possible factors of this equation
This will be done using the rational root theorem, which states that the factors of
f
(
x
)
=
p
x
n
+
q
x
m
+
...
+
C
are given by
Factors of C
Factors of p
, where
p
is the coefficient of the highest power of
x
.
Therefore, our possible factors are given by
±
1
,
±
2
,
±
4
,
±
5
,
±
10
,
±
20
±
2
,
±
1
Step 2: Check these factors using the remainder theorem
This is when the trial and error process comes into play. The remainder theorem is useful because you can check the remainders without actually having to do the division.
It states that if
x
−
a
is a factor of
f
(
x
)
, then
f
(
a
)
=
0
. With a little bit of trial and error, we find that
x
=
−
4
is a root because if
f
(
x
)
=
2
x
3
+
2
x
2
−
19
x
+
20
, then
f
(
−
4
)
=
2
(
−
4
)
3
+
2
(
−
4
)
2
−
19
(
−
4
)
+
20
=
0
.
Step 3: Use long division to simplify the equation
We have now found one root. We're not quite certain of the others yet. We can now simplify the equation to a quadratic by using long division or synthetic division.