What is the maximum number of zeros for the following function
f(x) = x4 + x3 + 7x2 + 9x - 18

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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