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Example.
If
f
(
x
)=
x
3
+2
x
2
+
x

1and
g
(
x
)=
x
2
+1,then
f
(
x
)
g
(
x
)
=
x
+2

3
x
2
+1
=
q
(
x
)+
r
(
x
)
g
(
x
)
where
q
(
x
)=
x
+ 2 (Quotient)
r
(
x
)=

3 (Remainder)
In general if
f
(
x
)and
g
(
x
) are polynomials, then
f
(
x
)
g
(
x
)
=
q
(
x
)+
r
(
x
)
g
(
x
)
where
r
(
x
) is a polynomial of degree less than the degree of
g
(
x
).
If
g
(
x
)=
x

c
in the above, then
f
(
x
)
x

c
=
q
(
x
)+
r
(
x
)
x

c
and so
f
(
x
)=
q
(
x
)(
x

c
)+
r
(
x
)
Setting
x
=
c
in the last expression above gives
f
(
c
)=
r
(
c
). The above proves the following theorem.
The Remainder Theorem.
If
f
(
x
) is a polynomial, then the remainder of
f
(
x
)
x

c
is
f
(
c
).
The following examples illustrate the Remainder Theorem.
Example.
x
3

2
x
2
+3
x

1
x

3
has remainder of 17 because
x
3

2
x
2
+3
x

1=17when
x
=3.
Example.
x
3

2
x
2
+3
x

1
x
+3
has remainder of 55 because
x
3

2
x
2
+3
x

1=

55 when
x
=

3.
The Remainder Theorem implies the following theorem.
The Factor Theorem.
The term (
x

c
) is a factor of a polynomial
f
(
x
) iﬀ (iﬀ means if and only
if)
f
(
c
)=0.
The following examples illustrate the Factor Theorem.
Example.
x
3

2
x
2
+3
x

1
6
=0when
x
= 3. Hence (
x

3) is not a factor of
x
3

2
x
2
+3
x

1.
Example.
x
3

2
x
2
+3
x

1
6
=0when
x
=

3. Hence (
x
+ 3) is not a factor of
x
3

2
x
2
+3
x

1.
Example.
x
2

9=0when
x
= 3. Hence (
x

3) is a factor of
x
2

9.
Example.
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