Question

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

Answer 1

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
.