Write the function in factored form, and give all the possible rational zeros of the function.
f(x) = x^4 − x^3 + 7x^2 − 9x − 18
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Answer:
Step-by-step explanation:Explanation:
First notice that by reversing the signs of the coefficients of the terms with odd degree, the sum is zero. So
x
=
−
1
is a zero:
f
(
−
1
)
=
1
+
1
+
7
+
9
−
18
=
0
and
(
x
+
1
)
is a factor:
x
4
−
x
3
+
7
x
2
−
9
x
−
18
=
(
x
+
1
)
(
x
3
−
2
x
2
+
9
x
−
18
)
then factor by grouping...
=
(
x
+
1
)
(
(
x
3
−
2
x
2
)
+
(
9
x
−
18
)
)
=
(
x
+
1
)
(
x
2
(
x
−
2
)
+
9
(
x
−
2
)
)
=
(
x
+
1
)
(
x
2
+
9
)
(
x
−
2
)
then take square root of
−
9
to find:
=
(
x
+
1
)
(
x
−
3
i
)
(
x
+
3
i
)
(
x
−
2
)
So the zeros are
x
=
−
1
,
x
=
3
i
,
x
=
−
3
i
and
x
=
2
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