If polynomial function vanishes at
x
=
a
, then
(
x
−
a
)
must be one of its factors.
Thus
(
x
−
7
)
,
(
x
−
4
−
√
6
)
and
(
x
−
4
+
√
6
)
are factors of the required polynomial.
The simplest polynomial that satisfies this is
(
x
−
7
)
(
x
−
4
−
√
6
)
(
x
−
4
+
√
6
)
=
(
x
−
7
)
{
(
x
−
4
)
−
√
6
}
{
(
x
−
4
)
+
√
6
}
=
(
x
−
7
)
{
(
x
−
4
)
2
−
6
}
=
(
x
−
7
)
(
x
2
−
8
x
+
16
−
6
)
=
(
x
−
7
)
(
x
2
−
8
x
+
10
)
