EduSahara™ Assignment
Name : Factor Theorem
Chapter : Polynomials
Grade : CBSE Grade IX
License : Non Commercial Use
Question
1
1.
Find the value of
k
such that
x
3
−
3
x
2
+
k
x
+
60
is exactly divisible by
(
x
−
6
)
(i)
-27
(ii)
-26
(iii)
-31
(iv)
-28
(v)
-29
Question
2
2.
If
2
and
-2
are the zeros of the polynomial
f
(
x
)
=
a
x
4
+
12
x
3
+
b
x
2
−
48
x
−
32
, find the value of
a
and
b
(i)
-7
,
5
(ii)
-9
,
3
(iii)
4
,
-7
(iv)
4
,
-8
(v)
5
,
-8
Question
3
3.
Find the value of
a
and
b
such that
b
x
4
+
14
x
3
+
a
x
2
−
26
x
+
20
is exactly divisible by
(
2
x
2
+
9
x
−
5
)
(i)
-36
,
4
(ii)
3
,
-37
(iii)
-35
,
4
(iv)
-36
,
5
(v)
5
,
-35
Question
4
4.
If the polynomial
f
(
x
)
=
3
x
2
+
k
x
+
30
is exactly divisible by
(
3
x
+
6
)
,
find
k
(i)
19
(ii)
24
(iii)
21
(iv)
20
(v)
22
Question
5
5.
If the polynomials
−
2
x
2
+
a
x
+
33
and
a
x
2
−
6
x
−
27
leave the same remainder when divided by
(
x
+
3
)
,
find the value of
a
(i)
2
(ii)
3
(iii)
(-1)
(iv)
1
(v)
4
Question
6
6.
Which of the following are true?
a)
If (x + a) is a factor of f(x), then f(a) = 0
b)
Zero of a polynomial and zero polynomial are synonymous
c)
A polynomial of degree n has atmost n zeros
d)
Zero of a polynomial is the value of the variable for which the polynomial value is zero
e)
Zero of a polynomial and root of the polynomial are synonymous
f)
A linear polynomial in one variable has only one root
g)
If (x - a) is a factor of f(x), then f(a) = 0
(i)
{a,c}
(ii)
{c,d,e,f,g}
(iii)
{a,f,g}
(iv)
{b,d}
(v)
{a,b,e}
Question
7
7.
If
(
x
2
−
1
)
is a factor of
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
,
which of the following are true ?
a)
a + b + c = 0
b)
a + b + c = d + e
c)
b + d = 0
d)
d + e = 0
e)
a + b + c + d + e = 0
f)
a + c + e = 0
(i)
{d,a,f}
(ii)
{a,c}
(iii)
{c,e,f}
(iv)
{b,c,e}
(v)
{b,e}
Question
8
8.
Which of the following are true ?
a)
Division of a polynomial with another polynomial stops when the degree of the remainder equals the degree of the divisor
b)
If the degree of p(x) is less then the degree of d(x), we should not divide p(x) with d(x)
c)
If p(a) = 0, then (x + a) perfectly divides p(x)
d)
If p(x) is divided by (x - a), the remainder is p(a)
(i)
{a,d,b}
(ii)
{c,d}
(iii)
{a,c,b}
(iv)
{a,b}
(v)
{b,d}
Question
9
9.
Which of the following are possible values
for the length and breadth of a rectangle
whose area is
(
−
12
x
2
+
34
x
−
10
)
(i)
(
−
2
x
−
5
)
(
6
x
+
2
)
(ii)
(
−
2
x
+
5
)
(
6
x
+
2
)
(iii)
(
−
2
x
−
5
)
(
6
x
−
2
)
(iv)
(
−
3
x
−
4
)
(
6
x
+
2
)
(v)
(
−
2
x
+
5
)
(
6
x
−
2
)
Question
10
10.
In which of the cases,
g
(
x
)
is a factor of
f
(
x
)
?
(i)
f
(
x
)
=
(
x
3
−
3
x
2
−
x
+
3
)
,
g
(
x
)
=
(
−
x
+
6
)
(ii)
f
(
x
)
=
(
−
x
3
+
10
x
2
−
27
x
+
18
)
,
g
(
x
)
=
(
x
+
2
)
(iii)
f
(
x
)
=
(
−
3
x
3
+
10
x
2
+
43
x
+
30
)
,
g
(
x
)
=
(
3
x
+
5
)
(iv)
f
(
x
)
=
(
3
x
3
−
22
x
2
+
9
x
+
90
)
,
g
(
x
)
=
(
−
x
+
1
)
(v)
f
(
x
)
=
(
x
3
−
2
x
2
−
5
x
+
6
)
,
g
(
x
)
=
(
x
+
1
)
Question
11
11.
Which of the following polynomials is a multiple of
(
2
x
+
1
)
?
(i)
(
2
x
3
−
x
2
−
5
x
−
2
)
(ii)
(
2
x
3
+
3
x
2
−
14
x
−
15
)
(iii)
(
x
3
+
2
x
2
−
5
x
−
6
)
(iv)
(
2
x
3
−
5
x
2
−
8
x
+
20
)
(v)
(
2
x
3
+
5
x
2
−
13
x
−
30
)
Question
12
12.
Which of the following polynomials has
(
x
+
2
)
as a factor ?
(i)
(
6
x
3
+
27
x
2
+
3
x
−
36
)
(ii)
(
18
x
3
−
69
x
2
+
87
x
−
36
)
(iii)
(
4
x
3
+
8
x
2
−
9
x
−
18
)
(iv)
(
4
x
3
+
16
x
2
−
9
x
−
36
)
(v)
(
9
x
3
+
15
x
2
−
72
x
+
48
)
Question
13
13.
If
f
(
x
)
=
(
4
x
3
+
12
x
2
−
16
x
−
48
)
and
g
(
x
)
=
(
6
x
3
−
22
x
2
−
28
x
+
80
)
have a common factor, find the common factor
(i)
(
x
−
4
)
(ii)
(
2
x
+
4
)
(iii)
(
2
x
−
4
)
(iv)
(
3
x
−
5
)
(v)
(
x
+
3
)
Question
14
14.
Which of the following polynomials is not a multiple of
(
x
−
1
)
?
(i)
(
x
2
−
1
)
(ii)
(
3
x
2
−
x
−
2
)
(iii)
(
2
x
2
+
3
x
−
5
)
(iv)
(
3
x
2
+
5
x
+
2
)
(v)
(
3
x
2
+
x
−
4
)
Assignment Key
1) (iv)
2) (iv)
3) (i)
4) (iii)
5) (i)
6) (ii)
7) (iii)
8) (v)
9) (v)
10) (iii)
11) (i)
12) (iii)
13) (ii)
14) (iv)
