Analyze the product x4 + x2 + 1?
Answers
Answer:
To find this, first notice that
x
4
+
x
2
+
1
>
0
for all (real) values of
x
. So there are no linear factors, only quadratic ones.
x
4
+
x
2
+
1
=
(
a
x
2
+
b
x
+
c
)
(
d
x
2
+
e
x
+
f
)
Without bothering to multiply this out fully just yet, notice that the coefficient of
x
4
gives us
a
d
=
1
. We might as well let
a
=
1
and
d
=
1
.
...
=
(
x
2
+
b
x
+
c
)
(
x
2
+
e
x
+
f
)
Next, the coefficient of
x
3
gives us
b
+
e
=
0
, so
e
=
−
b
.
...
=
(
x
2
+
b
x
+
c
)
(
x
2
−
b
x
+
f
)
The constant term gives us
c
f
=
1
, so either
c
=
f
=
1
or
c
=
f
=
−
1
. Let's try
c
=
f
=
1
.
...
=
(
x
2
+
b
x
+
1
)
(
x
2
−
b
x
+
1
)
Note that the coefficient of
x
will vanish nicely when these are multiplied out.
Finally notice that the coefficient of
x
2
is
(
1
−
b
2
+
1
)
=
2
−
b
2
, giving us
1
=
2
−
b
2
, thus
b
2
=
1
, so
b
=
1
or
b
=
−
1
.
Answer:
(x2+x+1) (x2_x+1) is the product of x4+x2+1