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
+
48
is exactly divisible by
(
x
+
4
)
(i)
-16
(ii)
-17
(iii)
-18
(iv)
-15
(v)
-13
Question
2
2.
If
3
and
(
-1
2
)
are the zeros of the polynomial
f
(
x
)
=
2
x
4
−
5
x
3
+
a
x
2
+
125
x
+
b
, find the value of
a
and
b
(i)
-53
,
75
(ii)
76
,
-52
(iii)
-53
,
76
(iv)
74
,
-54
(v)
-52
,
75
Question
3
3.
Find the value of
a
and
b
such that
a
x
4
+
b
x
3
−
8
x
2
+
54
x
−
36
is exactly divisible by
(
2
x
2
+
2
x
−
4
)
(i)
-15
,
3
(ii)
-13
,
5
(iii)
5
,
-14
(iv)
4
,
-14
(v)
4
,
-13
Question
4
4.
If the polynomial
f
(
x
)
=
6
x
2
−
13
x
+
k
is exactly divisible by
(
2
x
−
1
)
,
find
k
(i)
2
(ii)
4
(iii)
5
(iv)
8
(v)
6
Question
5
5.
If the polynomials
a
x
2
−
5
x
+
36
and
5
x
2
+
a
x
−
66
leave the same remainder when divided by
(
x
+
3
)
,
find the value of
a
(i)
(-9)
(ii)
(-4)
(iii)
(-6)
(iv)
(-7)
(v)
(-5)
Question
6
6.
Which of the following are true?
a)
A polynomial of degree n has atmost n zeros
b)
Zero of a polynomial and zero polynomial are synonymous
c)
If (x - a) is a factor of f(x), then f(a) = 0
d)
A linear polynomial in one variable has only one root
e)
If (x + a) is a factor of f(x), then f(a) = 0
f)
Zero of a polynomial and root of the polynomial are synonymous
g)
Zero of a polynomial is the value of the variable for which the polynomial value is zero
(i)
{b,e,d}
(ii)
{e,c}
(iii)
{b,a}
(iv)
{b,f,g}
(v)
{a,c,d,f,g}
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 = 0
c)
d + e = 0
d)
b + d = 0
e)
a + b + c = d + e
f)
a + c + e = 0
(i)
{b,d,f}
(ii)
{c,b,d}
(iii)
{c,d}
(iv)
{e,a,f}
(v)
{a,b}
Question
8
8.
Which of the following are true ?
a)
If p(a) = 0, then (x + a) perfectly divides p(x)
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)
Division of a polynomial with another polynomial stops when the degree of the remainder equals the degree of the divisor
d)
If p(x) is divided by (x - a), the remainder is p(a)
(i)
{a,c,b}
(ii)
{a,b}
(iii)
{a,d,b}
(iv)
{c,d}
(v)
{b,d}
Question
9
9.
Which of the following are possible values
for the length and breadth of a rectangle
whose area is
(
6
x
2
+
7
x
+
1
)
(i)
(
−
6
x
−
1
)
(
−
x
−
1
)
(ii)
(
−
6
x
+
1
)
(
−
x
+
1
)
(iii)
(
x
−
5
)
(
−
x
+
1
)
(iv)
(
−
6
x
−
1
)
(
−
x
+
1
)
(v)
(
−
6
x
+
1
)
(
−
x
−
1
)
Question
10
10.
In which of the cases,
g
(
x
)
is a factor of
f
(
x
)
?
(i)
f
(
x
)
=
(
3
x
3
−
5
x
2
−
3
x
+
5
)
,
g
(
x
)
=
(
x
+
2
)
(ii)
f
(
x
)
=
(
3
x
3
+
26
x
2
+
63
x
+
40
)
,
g
(
x
)
=
(
−
3
x
+
5
)
(iii)
f
(
x
)
=
(
3
x
3
+
29
x
2
+
86
x
+
80
)
,
g
(
x
)
=
(
x
+
5
)
(iv)
f
(
x
)
=
(
−
9
x
3
−
18
x
2
+
31
x
+
40
)
,
g
(
x
)
=
(
−
x
+
1
)
(v)
f
(
x
)
=
(
−
3
x
3
−
4
x
2
+
9
x
+
10
)
,
g
(
x
)
=
(
3
x
+
8
)
Question
11
11.
Which of the following polynomials is a multiple of
(
x
−
2
)
?
(i)
(
6
x
3
+
5
x
2
−
3
x
−
2
)
(ii)
(
6
x
3
−
5
x
2
−
8
x
+
3
)
(iii)
(
6
x
3
−
25
x
2
+
32
x
−
12
)
(iv)
(
18
x
3
−
45
x
2
+
31
x
−
6
)
(v)
(
6
x
3
+
7
x
2
−
1
)
Question
12
12.
Which of the following polynomials has
(
2
x
+
4
)
as a factor ?
(i)
(
4
x
3
+
2
x
2
−
14
x
−
12
)
(ii)
(
4
x
3
−
26
x
2
+
56
x
−
40
)
(iii)
(
8
x
3
−
24
x
2
−
14
x
+
60
)
(iv)
(
8
x
3
+
36
x
2
+
52
x
+
24
)
(v)
(
4
x
3
−
12
x
2
+
16
)
Question
13
13.
If
f
(
x
)
=
(
4
x
3
+
8
x
2
−
15
x
−
9
)
and
g
(
x
)
=
(
2
x
3
+
16
x
2
+
38
x
+
24
)
have a common factor, find the common factor
(i)
(
2
x
+
2
)
(ii)
(
2
x
−
3
)
(iii)
(
x
+
4
)
(iv)
(
x
+
3
)
(v)
(
2
x
+
1
)
Question
14
14.
Which of the following polynomials is not a multiple of
(
3
x
+
5
)
?
(i)
(
2
x
2
+
3
x
−
5
)
(ii)
(
3
x
2
+
2
x
−
5
)
(iii)
(
6
x
2
+
25
x
+
25
)
(iv)
(
3
x
2
+
17
x
+
20
)
(v)
(
9
x
2
+
27
x
+
20
)
Assignment Key
1) (i)
2) (i)
3) (iv)
4) (iii)
5) (iii)
6) (v)
7) (i)
8) (v)
9) (i)
10) (iii)
11) (iii)
12) (iv)
13) (iv)
14) (i)
