show that (x+a)is a factor ofx^n-a^n
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Answer:
According to factor theorem, if
x
−
k
is a factor of a polynomial function
f
(
x
)
, then
f
(
k
)
=
0
Now to test for
x
+
a
being a factor of
f
(
x
)
=
x
n
+
a
n
,
as
x
+
a
=
x
−
(
−
a
)
, we must check for
f
(
−
a
)
Now
f
(
x
)
=
x
n
+
a
n
Hence,
f
(
−
a
)
=
(
−
a
)
n
+
a
n
Observe if
n
is even then
f
(
−
a
)
=
(
−
a
)
n
+
a
n
=
a
n
+
a
n
=
2
a
n
≠
0
and hence
x
+
a
is not a factor of
x
n
+
a
n
if
n
is even
but if
n
is odd then
f
(
−
a
)
=
(
−
a
)
n
+
a
n
=
−
a
n
+
a
n
=
0
and hence
x
+
a
is a factor of
x
n
+
a
n
if
n
is odd.
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