Graph

A third degree polynomial with 2 zeros

For a third degree polynomial, we need 3 linear factors.

Since

5

and

2

i

are roots (zeros), we know that

x

-

5

and

x

-

2

i

are factors.

If we want a polynomial with real coeficients, then the complex conjugate of

2

i

(which is

-

2

i

) must also be a root and

x

+

2

i

must be a factor.

One polynomial with real coefficients that meets the requirements is

(

x

-

5

)

(

x

-

2

i

)

(

x

+

2

i

)

=

(

x

-

5

)

(

x

2

+

4

)

=

x

3

-

5

x

2

+

4

x

-

20

Any constant multiple of this also meets the requirements.

For example

7

(

x

3

-

5

x

2

+

4

x

-

20

)

=

7

x

3

-

35

x

2

+

28

x

-

140