What is a possible factorization of a polynomial with - 5, 3, and i as zeros?

The best way is to recognise that, if

x

=

5

is a root, then

x

-

5

=

0

, and ditto for the other two roots. So we have

x

-

5

,

x

-

i

,

x

+

i

all equalling zero. To find our polynomial, we just multiply the three terms together:

(

x

-

5

)

(

x

-

i

)

(

x

+

i

)

=

(

x

2

-

i

x

-

5

x

+

5

i

)

(

x

+

i

)

=

x

3

+

i

x

2

-

i

x

2

-

(

i

2

)

x

-

5

x

2

-

5

i

x

+

5

i

x

+

5

i

2

which simplifies to

x

3

-

5

x

2

+

x

-

5

.