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Answered by
1
Answer:
when x^1999 is divided by x^2-1 then it leaves remainder as x.
Answered by
2
Answer:
Step-by-step explanation:
Note that:
(
x
2
−
1
)
(
x
1997
+
x
1995
+
x
1993
+
...
+
x
3
+
x
)
=
x
2
(
x
1997
+
x
1995
+
x
1993
+
...
+
x
3
+
x
)
−
(
x
1997
+
x
1995
+
x
1993
+
...
+
x
3
+
x
)
=
(
x
1999
+
x
1997
+
x
1995
+
...
+
x
5
+
x
3
)
−
(
x
1997
+
x
1995
+
x
1993
+
...
+
x
3
+
x
)
=
x
1999
−
x
So:
x
1999
=
(
x
2
−
1
)
(
x
1997
+
x
1995
+
x
1993
+
...
+
x
3
+
x
)
+
x
That is:
x
1999
x
2
−
1
=
x
1997
+
x
1995
+
x
1993
+
...
+
x
3
+
x
with remainder
x
The polynomial
x
is of degree less than the degree of
x
2
−
1
, so is the remainder we want
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