Answer:
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
