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 About the author Serenity
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 Reply
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