EduSahara™ Assignment
Name : Factor Theorem
Chapter : Factor Theorem
Grade : ICSE Grade X
License : Non Commercial Use
Question
1
1.
The remainder when
g
is divided by
3
is
(i)
1
(ii)
0
(iii)
(
−
1
)
(iv)
3
(v)
(
−
3
)
Question
2
2.
The remainder when
e
2
is divided by
(
e
−
5
)
is
(i)
24
(ii)
25
(iii)
26
(iv)
28
(v)
22
Question
3
3.
The remainder when
(
6
n
+
6
)
is divided by
(
n
−
8
)
is
(i)
53
(ii)
51
(iii)
54
(iv)
55
(v)
56
Question
4
4.
The remainder when
(
2
r
2
−
2
r
)
is divided by
(
r
+
4
)
is
(i)
40
(ii)
43
(iii)
37
(iv)
41
(v)
39
Question
5
5.
The remainder when
(
−
4
s
2
+
9
s
+
2
)
is divided by
(
s
+
8
)
is
(i)
(
−
328
)
(ii)
(
−
324
)
(iii)
(
−
327
)
(iv)
(
−
326
)
(v)
(
−
325
)
Question
6
6.
The remainder when
(
−
2
r
4
+
2
r
3
−
4
r
2
−
9
r
+
7
)
is divided by
(
r
2
+
8
r
+
12
)
is
(i)
(
768
r
+
1495
)
(ii)
(
767
r
+
1495
)
(iii)
(
765
r
+
1495
)
(iv)
(
766
r
+
1495
)
(v)
(
769
r
+
1495
)
Question
7
7.
Factorize
(
6
x
2
−
25
x
+
14
)
(i)
(
2
x
−
7
)
(
2
x
−
2
)
(ii)
(
2
x
−
7
)
(
4
x
−
2
)
(iii)
(
2
x
−
7
)
(
6
x
−
2
)
(iv)
(
2
x
−
7
)
(
−
2
)
(v)
(
2
x
−
7
)
(
3
x
−
2
)
Question
8
8.
Factorize
(
2
x
3
+
16
x
2
−
58
x
−
360
)
(i)
(
2
x
+
8
)
(
−
2
x
+
9
)
(
−
2
x
−
5
)
(ii)
(
2
x
+
8
)
9
(
−
5
)
(iii)
(
2
x
+
8
)
(
x
+
9
)
(
x
−
5
)
(iv)
(
2
x
+
8
)
(
2
x
+
9
)
(
2
x
−
5
)
(v)
(
2
x
+
8
)
(
4
x
+
9
)
(
4
x
−
5
)
Question
9
9.
Factorize
(
x
4
+
22
x
3
+
171
x
2
+
542
x
+
560
)
(i)
(
x
+
2
)
(
−
2
x
+
7
)
(
−
2
x
+
8
)
(
−
x
+
5
)
(ii)
(
x
+
2
)
(
2
x
+
7
)
(
2
x
+
8
)
(
2
x
+
5
)
(iii)
(
x
+
2
)
(
3
x
+
7
)
(
4
x
+
8
)
(
4
x
+
5
)
(iv)
(
x
+
2
)
7
8
5
(v)
(
x
+
2
)
(
x
+
7
)
(
x
+
8
)
(
x
+
5
)
Question
10
10.
The value of the polynomial
(
−
6
k
+
1
)
at
k
=
3
is
(i)
-18
(ii)
-16
(iii)
-17
(iv)
-19
(v)
-15
Question
11
11.
The value of the polynomial
(
8
p
2
−
6
p
−
2
)
at
p
=
(-4)
is
(i)
152
(ii)
148
(iii)
150
(iv)
149
(v)
151
Question
12
12.
The value of the polynomial
(
−
8
f
3
+
6
f
+
5
)
at
f
=
0
is
(i)
4
(ii)
7
(iii)
6
(iv)
3
(v)
5
Question
13
13.
The value of the polynomial
(
9
t
4
+
3
t
3
−
t
2
−
7
t
−
7
)
at
t
=
(-2)
is
(i)
125
(ii)
122
(iii)
120
(iv)
124
(v)
123
Question
14
14.
Factorize
(
x
2
+
2
x
a
+
3
x
+
a
2
+
3
a
−
28
)
(i)
(
x
+
a
+
7
)
(
x
+
a
−
4
)
(ii)
(
x
−
a
+
7
)
(
x
+
a
+
4
)
(iii)
(
x
+
a
−
4
)
(
x
−
a
+
7
)
(iv)
(
x
+
a
+
7
)
(
x
−
a
−
4
)
(v)
(
x
−
a
−
4
)
(
x
+
a
−
7
)
Question
15
15.
Factorize
(
x
3
+
3
x
2
a
+
3
x
a
2
−
x
+
a
3
−
a
)
(i)
(
x
+
a
)
(
x
−
a
−
1
)
(
x
+
a
+
1
)
(ii)
(
x
+
a
)
(
x
+
a
−
1
)
(
x
−
a
+
1
)
(iii)
(
x
−
a
)
(
x
+
a
−
1
)
(
x
−
a
+
1
)
(iv)
(
x
+
a
)
(
x
+
a
−
1
)
(
x
+
a
+
1
)
(v)
(
x
−
a
)
(
x
−
a
−
1
)
(
x
+
a
+
1
)
Question
16
16.
Factorize
(
9
x
2
+
25
x
y
+
14
y
2
)
(i)
(
x
+
2
y
)
(
9
x
+
7
y
)
(ii)
(
x
+
2
y
)
(
x
−
2
y
)
(iii)
(
x
−
2
y
)
(
9
x
−
7
y
)
(iv)
(
x
−
2
y
)
(
9
x
+
7
y
)
(v)
(
x
+
2
y
)
(
9
x
−
7
y
)
Question
17
17.
Find the value of
k
such that
k
x
4
−
80
x
3
+
266
x
2
−
366
x
+
180
is exactly divisible by
(
2
x
−
4
)
(i)
8
(ii)
9
(iii)
7
(iv)
10
(v)
6
Question
18
18.
If
1
and
(
-3
2
)
are the zeros of the polynomial
f
(
x
)
=
4
x
4
+
30
x
3
+
48
x
2
+
a
x
+
b
, find the value of
a
and
b
(i)
-22
,
-59
(ii)
-61
,
-23
(iii)
-59
,
-21
(iv)
-22
,
-60
(v)
-21
,
-60
Question
19
19.
Find the value of
a
and
b
such that
a
x
4
−
4
x
3
−
42
x
2
+
b
x
+
54
is exactly divisible by
(
4
x
2
−
9
)
(i)
8
,
7
(ii)
10
,
9
(iii)
9
,
9
(iv)
8
,
9
(v)
8
,
10
Question
20
20.
If
4
is the zero of the polynomial
f
(
x
)
=
x
2
+
k
x
+
16
,
find
k
(i)
-9
(ii)
-7
(iii)
-8
(iv)
-5
(v)
-10
Question
21
21.
If the polynomial
f
(
x
)
=
k
x
2
+
17
x
−
6
is exactly divisible by
(
x
+
6
)
,
find
k
(i)
4
(ii)
0
(iii)
6
(iv)
2
(v)
3
Question
22
22.
If the polynomial
a
x
4
+
5
x
3
−
14
x
2
+
b
x
−
30
is divided by
(
x
−
2
)
,
it leaves a remainder
-108
.
If it is divided by
(
x
−
6
)
,
it leaves a remainder
2856
.
Find the value of
a
and
b
(i)
2
,
-46
(ii)
3
,
-47
(iii)
-48
,
1
(iv)
-46
,
3
(v)
2
,
-47
Question
23
23.
If the polynomials
a
x
2
+
6
x
−
23
and
−
x
2
+
a
x
+
10
leave the same remainder when divided by
(
x
−
3
)
,
find the value of
a
(i)
1
(ii)
4
(iii)
0
(iv)
2
(v)
(-1)
Question
24
24.
Which of the following are true?
a)
Zero of a polynomial and root of the polynomial are synonymous
b)
Zero of a polynomial is the value of the variable for which the polynomial value is zero
c)
If (x - a) is a factor of f(x), then f(a) = 0
d)
If (x + a) is a factor of f(x), then f(a) = 0
e)
A polynomial of degree n has atmost n zeros
f)
Zero of a polynomial and zero polynomial are synonymous
g)
A linear polynomial in one variable has only one root
(i)
{d,f,c}
(ii)
{f,b}
(iii)
{d,e,g}
(iv)
{d,a}
(v)
{a,b,c,e,g}
Question
25
25.
Which of the following are true?
a)
A binomial may have degree 3
b)
Every polynomial is a binomial
c)
A binomial has two and only two terms
d)
π
r
2
is a monomial
e)
Degree of zero polynomial is zero
(i)
{b,a}
(ii)
{e,c}
(iii)
{b,a,c}
(iv)
{a,c,d}
(v)
{b,e,d}
Question
26
26.
Given
f
(
z
)
=
(
−
3
z
−
6
)
,
find
f
(
1
)
(i)
-8
(ii)
-10
(iii)
-12
(iv)
-6
(v)
-9
Question
27
27.
Given
f
(
v
)
=
(
8
v
2
+
7
v
+
4
)
,
find
f
(
2
)
(i)
48
(ii)
50
(iii)
51
(iv)
52
(v)
49
Question
28
28.
Given
f
(
r
)
=
(
−
8
r
3
+
8
r
2
−
5
)
,
find
f
(
(-3)
)
(i)
282
(ii)
285
(iii)
280
(iv)
283
(v)
284
Question
29
29.
Given
f
(
j
)
=
(
−
4
j
4
−
7
j
3
−
2
j
2
−
4
j
−
3
)
,
find
f
(
3
)
(i)
-546
(ii)
-547
(iii)
-544
(iv)
-545
(v)
-549
Question
30
30.
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)
b + d = 0
b)
a + b + c + d + e = 0
c)
a + b + c = d + e
d)
d + e = 0
e)
a + c + e = 0
f)
a + b + c = 0
(i)
{f,c,e}
(ii)
{d,b}
(iii)
{d,a,b}
(iv)
{a,b,e}
(v)
{c,a}
Question
31
31.
Which of the following are true ?
a)
If p(x) is divided by (x - a), the remainder is p(a)
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)
Division of a polynomial with another polynomial stops when the degree of the remainder equals the degree of the divisor
(i)
{c,a}
(ii)
{c,d,a}
(iii)
{a,b}
(iv)
{c,b,a}
(v)
{d,b}
Question
32
32.
Which of the following are possible values
for the length and breadth of a rectangle
whose area is
(
20
x
2
−
8
x
−
12
)
(i)
(
−
4
x
+
4
)
(
−
5
x
+
3
)
(ii)
(
−
4
x
−
4
)
(
−
5
x
−
3
)
(iii)
(
5
x
−
5
)
(
−
5
x
+
3
)
(iv)
(
−
4
x
−
4
)
(
−
5
x
+
3
)
(v)
(
−
4
x
+
4
)
(
−
5
x
−
3
)
Question
33
33.
In which of the cases,
g
(
x
)
is a factor of
f
(
x
)
?
(i)
f
(
x
)
=
(
−
3
x
3
−
10
x
2
+
27
x
+
10
)
,
g
(
x
)
=
(
−
x
+
2
)
(ii)
f
(
x
)
=
(
−
3
x
3
−
x
2
+
12
x
+
4
)
,
g
(
x
)
=
(
2
x
+
5
)
(iii)
f
(
x
)
=
(
2
x
3
+
19
x
2
+
55
x
+
50
)
,
g
(
x
)
=
(
3
x
+
1
)
(iv)
f
(
x
)
=
(
6
x
3
+
35
x
2
+
66
x
+
40
)
,
g
(
x
)
=
(
x
+
5
)
(v)
f
(
x
)
=
(
6
x
3
+
29
x
2
+
39
x
+
10
)
,
g
(
x
)
=
(
3
x
+
4
)
Question
34
34.
Which of the following polynomials is a multiple of
(
x
+
1
)
?
(i)
(
2
x
3
+
3
x
2
−
9
x
−
10
)
(ii)
(
6
x
3
−
25
x
2
+
29
x
−
10
)
(iii)
(
6
x
3
−
31
x
2
+
48
x
−
20
)
(iv)
(
6
x
3
+
5
x
2
−
21
x
+
10
)
(v)
(
4
x
3
−
8
x
2
−
25
x
+
50
)
Question
35
35.
Which of the following polynomials has
(
3
x
−
1
)
as a factor ?
(i)
(
9
x
3
−
60
x
2
+
79
x
−
20
)
(ii)
(
3
x
3
−
25
x
2
+
58
x
−
40
)
(iii)
(
6
x
3
−
23
x
2
+
26
x
−
8
)
(iv)
(
6
x
3
−
19
x
2
+
16
x
−
4
)
(v)
(
6
x
3
−
37
x
2
+
37
x
−
10
)
Question
36
36.
Find the remainder when
(
4
x
2
−
18
x
+
20
)
is divided by
(
x
−
8
)
(i)
129
(ii)
134
(iii)
131
(iv)
132
(v)
133
Question
37
37.
If
f
(
x
)
=
(
3
x
3
−
19
x
2
+
38
x
−
24
)
and
g
(
x
)
=
(
6
x
3
−
25
x
2
+
23
x
−
6
)
have a common factor, find the common factor
(i)
(
3
x
−
4
)
(ii)
(
x
−
2
)
(iii)
(
3
x
−
2
)
(iv)
(
x
−
3
)
(v)
(
2
x
−
1
)
Question
38
38.
Find the quadratic polynomial which when divided by
(
x
−
1
)
,
(
3
x
−
6
)
,
(
3
x
+
4
)
leaves remainders of
8
,
1
,
143
3
respectively.
(i)
(
3
x
2
−
5
x
−
12
)
(ii)
(
3
x
2
−
9
x
+
6
)
(iii)
(
3
x
2
−
10
x
+
7
)
(iv)
(
3
x
2
−
16
x
+
21
)
(v)
(
9
x
2
−
6
x
−
24
)
Question
39
39.
Which of the following polynomials is not a multiple of
(
x
+
3
)
?
(i)
(
3
x
2
−
2
x
−
5
)
(ii)
(
3
x
2
+
4
x
−
15
)
(iii)
(
x
2
+
2
x
−
3
)
(iv)
(
x
2
+
8
x
+
15
)
(v)
(
x
2
+
4
x
+
3
)
Question
40
40.
If the polynomial
b
x
2
+
a
x
−
12
is divided by
(
2
x
−
4
)
,
it leaves a remainder
10
.
If it is divided by
(
2
x
−
3
)
,
it leaves a remainder
2
1
4
.
Find the value of
a
and
b
(i)
5
,
4
(ii)
6
,
3
(iii)
2
,
4
(iv)
4
,
6
(v)
5
,
3
Assignment Key
1) (ii)
2) (ii)
3) (iii)
4) (i)
5) (iv)
6) (ii)
7) (v)
8) (iii)
9) (v)
10) (iii)
11) (iii)
12) (v)
13) (v)
14) (i)
15) (iv)
16) (i)
17) (i)
18) (iv)
19) (iv)
20) (iii)
21) (v)
22) (v)
23) (i)
24) (v)
25) (iv)
26) (v)
27) (ii)
28) (iv)
29) (i)
30) (iv)
31) (iii)
32) (v)
33) (i)
34) (i)
35) (i)
36) (iv)
37) (iv)
38) (iv)
39) (i)
40) (v)
