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Byju's Answer
Standard X
Mathematics
GCD of Polynomials
If x2-1 is ...
Question
If
x
2
−
1
is a factor of
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
, show that
a
+
c
+
e
=
b
+
d
=
0
.
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Solution
p
(
x
)
=
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
Then,
(
x
2
−
1
)
is a factor of
p
(
x
)
(
x
−
1
)
(
x
+
1
)
is a factor of
p
(
x
)
(
x
−
1
)
and
(
x
+
1
)
are factors of
p
(
x
)
p
(
1
)
=
0
and
p
(
−
1
)
=
0
a
+
b
+
c
+
d
+
e
=
0
and
a
−
b
+
c
−
d
+
e
=
0
Adding and subtracting these two equations, we get,
2
(
a
+
c
+
e
)
=
0
and
2
(
b
+
d
)
=
0
⟹
a
+
c
+
e
=
0
and
b
+
d
=
0
⟹
a
+
c
+
e
=
b
+
d
=
0
Suggest Corrections
2
Similar questions
Q.
If
(
x
2
−
1
)
is factor of
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
, show that
a
+
c
+
e
=
b
+
d
=
0
Q.
If
x
3
+
1
is a factor of
a
x
2
+
b
x
3
+
c
x
2
+
d
x
+
e
,
a, b, c, d, e
ϵ
R
a
n
d
a
≠
0
,
then the real root of
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
=
0
other than -1 is
Q.
If x
2
− 1 is a factor of ax
4
+ bx
3
+ cx
2
+ dx + e, then
(a) a + c + e = b + d
(b) a + b +e = c + d
(c) a + b + c = d + e
(d) b + c + d = a + e