EduSahara™ Assignment
Name : Problems based on Zeros of Polynomial
Chapter : Polynomials
Grade : CBSE Grade X
License : Non Commercial Use
Question
1
1.
Find the roots of the quadratic equation
(
x
2
−
12
x
+
27
)
=
0
(i)
(
12
,
2
)
(ii)
(
10
,
2
)
(iii)
(
9
,
3
)
(iv)
(
10
,
3
)
(v)
(
12
,
0
)
Question
2
2.
Find the roots of the quadratic equation
(
x
2
+
6
x
+
8
)
=
0
(i)
(
-2
,
-4
)
(ii)
(
-1
,
-4
)
(iii)
(
1
,
-6
)
(iv)
(
-1
,
-5
)
(v)
(
1
,
-5
)
Question
3
3.
Find the roots of the quadratic equation
(
12
x
2
−
22
x
+
10
)
=
0
(i)
(
4
,
1
2
)
(ii)
(
1
,
5
6
)
(iii)
(
2
,
5
4
)
(iv)
(
4
,
5
4
)
(v)
(
2
,
5
6
)
Question
4
4.
Find the roots of the quadratic equation
(
56
x
2
+
25
x
−
4
)
=
0
(i)
(
3
8
,
(
-6
7
)
)
(ii)
(
1
8
,
(
-4
7
)
)
(iii)
(
1
10
,
(
-4
7
)
)
(iv)
(
1
10
,
(
-4
5
)
)
(v)
(
3
8
,
(
-4
5
)
)
Question
5
5.
Find the quadratic equation with roots
(
8
3
,
8
3
)
(i)
(
3
x
2
−
32
x
+
64
)
=
0
(ii)
(
15
x
2
−
64
x
+
64
)
=
0
(iii)
(
9
x
2
−
54
x
+
80
)
=
0
(iv)
(
3
x
2
−
14
x
+
16
)
=
0
(v)
(
9
x
2
−
48
x
+
64
)
=
0
Question
6
6.
Find the quadratic equation with roots
(
8
9
,
5
9
)
(i)
(
99
x
2
−
133
x
+
40
)
=
0
(ii)
(
63
x
2
−
107
x
+
40
)
=
0
(iii)
(
81
x
2
−
117
x
+
40
)
=
0
(iv)
(
27
x
2
−
33
x
+
10
)
=
0
(v)
(
81
x
2
−
135
x
+
56
)
=
0
Question
7
7.
Find the quadratic equation with roots
(
-7
,
-7
)
(i)
(
x
2
+
12
x
+
35
)
=
0
(ii)
(
x
2
+
17
x
+
70
)
=
0
(iii)
(
x
2
+
13
x
+
42
)
=
0
(iv)
(
x
2
+
14
x
+
49
)
=
0
(v)
(
x
2
+
15
x
+
56
)
=
0
Question
8
8.
Find the quadratic equation with roots
(
4
,
-4
)
(i)
(
x
2
+
x
−
12
)
=
0
(ii)
(
x
2
−
16
)
=
0
(iii)
(
x
2
−
x
−
12
)
=
0
(iv)
(
x
2
−
3
x
−
4
)
=
0
(v)
(
x
2
+
3
x
−
4
)
=
0
Question
9
9.
The sum of the roots of the quadratic equation
(
x
2
+
12
x
+
36
)
=
0
is
(i)
-10
(ii)
-14
(iii)
-13
(iv)
-11
(v)
-12
Question
10
10.
The sum of the roots of the quadratic equation
(
x
2
+
6
x
+
5
)
=
0
is
(i)
-9
(ii)
-6
(iii)
-5
(iv)
-4
(v)
-7
Question
11
11.
The sum of the roots of the quadratic equation
(
64
x
2
+
96
x
+
36
)
=
0
is
(i)
(
-5
2
)
(ii)
(
-3
2
)
(iii)
-2
(iv)
(
-1
2
)
(v)
(
-5
4
)
Question
12
12.
The sum of the roots of the quadratic equation
(
−
45
x
2
+
72
x
)
=
0
is
(i)
8
5
(ii)
10
7
(iii)
2
(iv)
6
5
Question
13
13.
The product of the roots of the quadratic equation
(
x
2
−
14
x
+
49
)
=
0
is
(i)
48
(ii)
47
(iii)
50
(iv)
52
(v)
49
Question
14
14.
The product of the roots of the quadratic equation
(
x
2
+
11
x
+
30
)
=
0
is
(i)
30
(ii)
27
(iii)
29
(iv)
31
(v)
33
Question
15
15.
The product of the roots of the quadratic equation
(
64
x
2
+
64
x
+
16
)
=
0
is
(i)
1
2
(ii)
1
4
(iii)
(
-1
4
)
(iv)
1
6
(v)
3
4
Question
16
16.
The product of the roots of the quadratic equation
(
−
18
x
2
+
63
x
−
27
)
=
0
is
(i)
3
2
(ii)
2
(iii)
5
4
(iv)
5
2
(v)
1
2
Question
17
17.
Find the quadratic equation, the sum of whose roots is
-6
and product is
9
(i)
(
2
x
2
+
6
x
+
9
)
=
0
(ii)
(
x
2
+
9
x
+
9
)
=
0
(iii)
(
x
2
+
3
x
+
9
)
=
0
(iv)
(
x
2
+
6
x
+
9
)
=
0
(v)
(
6
x
+
9
)
=
0
Question
18
18.
Find the quadratic equation, the sum of whose roots is
14
and product is
45
(i)
(
x
2
−
14
x
+
45
)
=
0
(ii)
(
2
x
2
−
14
x
+
45
)
=
0
(iii)
(
x
2
−
12
x
+
45
)
=
0
(iv)
(
x
2
−
17
x
+
45
)
=
0
(v)
(
−
14
x
+
45
)
=
0
Question
19
19.
Find the quadratic equation, the sum of whose roots is
-9
and product is
81
4
(i)
(
4
x
2
+
36
x
+
81
)
=
0
(ii)
(
5
x
2
+
36
x
+
81
)
=
0
(iii)
(
4
x
2
+
33
x
+
81
)
=
0
(iv)
(
3
x
2
+
36
x
+
81
)
=
0
(v)
(
4
x
2
+
39
x
+
81
)
=
0
Question
20
20.
Find the quadratic equation, the sum of whose roots is
4
9
and product is
(
-5
81
)
(i)
(
−
81
x
2
+
36
x
+
5
)
=
0
(ii)
(
−
82
x
2
+
36
x
+
5
)
=
0
(iii)
(
−
81
x
2
+
39
x
+
5
)
=
0
(iv)
(
−
80
x
2
+
36
x
+
5
)
=
0
(v)
(
−
81
x
2
+
33
x
+
5
)
=
0
Question
21
21.
Find the roots of the quadratic equation
(
−
7
x
2
−
9
x
+
6
)
=
0
(i)
(
(
−
9
14
−
1
14
4
√
249
)
,
(
−
9
14
+
1
14
√
249
)
)
(ii)
(
(
−
9
14
−
1
14
4
√
249
)
,
(
−
9
14
+
249
14
)
)
(iii)
(
(
−
1
2
−
1
14
√
249
)
,
(
−
11
14
+
1
14
√
249
)
)
(iv)
(
(
−
9
14
−
1
14
√
249
)
,
(
−
9
14
+
1
14
√
249
)
)
(v)
(
(
−
1
2
−
1
14
√
249
)
,
(
−
9
14
+
249
14
)
)
Question
22
22.
Find the value of
k
such that
8
x
4
+
k
x
3
−
98
x
2
−
414
x
−
360
is exactly divisible by
(
x
−
4
)
(i)
26
(ii)
21
(iii)
24
(iv)
25
(v)
23
Question
23
23.
If
3
and
-1
are the zeros of the polynomial
f
(
x
)
=
2
x
4
−
6
x
3
+
b
x
2
+
a
x
+
36
, find the value of
a
and
b
(i)
30
,
-13
(ii)
30
,
-14
(iii)
-15
,
29
(iv)
-13
,
31
(v)
31
,
-14
Question
24
24.
Find the value of
a
and
b
such that
b
x
4
+
a
x
3
−
56
x
2
−
25
x
+
150
is exactly divisible by
(
x
2
−
25
)
(i)
1
,
0
(ii)
2
,
2
(iii)
1
,
2
(iv)
1
,
3
(v)
3
,
2
Question
25
25.
Factorize and divide
(
900
x
4
−
436
x
2
+
16
)
÷
(
30
x
2
+
26
x
+
4
)
(i)
(
31
x
2
−
26
x
+
4
)
(ii)
(
32
x
2
−
26
x
+
4
)
(iii)
(
29
x
2
−
26
x
+
4
)
(iv)
(
28
x
2
−
26
x
+
4
)
(v)
(
30
x
2
−
26
x
+
4
)
Question
26
26.
Factorize and divide
(
8
x
4
+
4
x
3
−
96
x
2
−
36
x
+
216
)
÷
(
−
x
2
−
5
x
−
6
)
(i)
(
−
8
x
2
−
36
x
−
36
)
(ii)
(
−
4
x
2
−
2
x
+
12
)
(iii)
(
−
8
x
2
+
12
x
+
36
)
(iv)
(
−
8
x
2
−
12
x
+
36
)
(v)
(
−
8
x
2
+
36
x
−
36
)
Question
27
27.
If
(
-1
2
)
is the zero of the polynomial
f
(
x
)
=
2
x
2
+
3
x
+
k
,
find
k
(i)
1
(ii)
-1
(iii)
4
(iv)
0
(v)
2
Question
28
28.
If the polynomial
f
(
x
)
=
9
x
2
−
3
x
+
k
is exactly divisible by
(
3
x
−
4
)
,
find
k
(i)
-12
(ii)
-11
(iii)
-10
(iv)
-15
(v)
-13
Question
29
29.
If the polynomial
2
x
4
−
5
x
3
+
a
x
2
+
b
x
+
54
is divided by
(
x
−
2
)
,
it leaves a remainder
-140
.
If it is divided by
(
x
−
1
)
,
it leaves a remainder
0
.
Find the value of
a
and
b
(i)
-8
,
-41
(ii)
-41
,
-9
(iii)
-42
,
-9
(iv)
-10
,
-43
(v)
-42
,
-8
Question
30
30.
If the polynomials
3
x
2
+
a
x
−
24
and
a
x
2
+
2
x
−
24
leave the same remainder when divided by
(
x
−
2
)
,
find the value of
a
(i)
6
(ii)
4
(iii)
3
(iv)
5
(v)
1
Question
31
31.
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 is the value of the variable for which the polynomial value is zero
c)
A linear polynomial in one variable has only one root
d)
Zero of a polynomial and root of the polynomial are synonymous
e)
Zero of a polynomial and zero polynomial are synonymous
f)
If (x + a) is a factor of f(x), then f(a) = 0
g)
A polynomial of degree n has atmost n zeros
(i)
{a,b,c,d,g}
(ii)
{e,a}
(iii)
{f,b}
(iv)
{e,f,c}
(v)
{e,d,g}
Question
32
32.
Which of the following are true?
a)
π
r
2
is a monomial
b)
A binomial may have degree 3
c)
Every polynomial is a binomial
d)
Degree of zero polynomial is zero
e)
A binomial has two and only two terms
(i)
{a,b,e}
(ii)
{c,a}
(iii)
{d,b}
(iv)
{c,d,e}
(v)
{c,a,b}
Question
33
33.
Which of the following is a factor of
37
x
5
y
2
z
5
?
(i)
37
x
5
z
6
(ii)
37
x
3
z
6
(iii)
37
x
3
z
3
(iv)
37
x
3
y
3
z
3
(v)
x
6
z
3
Question
34
34.
Which of the following is not a factor of
36
x
5
y
z
2
?
(i)
12
y
z
(ii)
12
x
6
y
2
z
3
(iii)
12
x
5
z
2
(iv)
12
x
5
y
z
(v)
12
x
4
y
z
2
Question
35
35.
Which of the following is a factor of
(
8
x
5
+
y
4
z
5
)
?
(i)
no factors
(ii)
y
4
z
5
(iii)
2
x
z
4
(iv)
8
x
5
(v)
2
y
4
Question
36
36.
Which of the following is an irreducible factor of
37
x
3
y
2
z
3
?
(i)
z
(ii)
x
3
y
2
z
2
(iii)
y
2
z
(iv)
x
z
2
(v)
x
3
y
Question
37
37.
Which of the following is not an irreducible factor of
(
x
2
y
+
x
y
2
+
x
y
)
?
(i)
x
y
(ii)
(
x
+
y
+
1
)
(iii)
y
(iv)
x
Question
38
38.
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)
b + d = 0
d)
d + e = 0
e)
a + b + c = d + e
f)
a + c + e = 0
(i)
{d,b,c}
(ii)
{e,a,f}
(iii)
{b,c,f}
(iv)
{d,c}
(v)
{a,b}
Question
39
39.
Which of the following are true ?
a)
If p(x) is divided by (x - a), the remainder is p(a)
b)
If p(a) = 0, then (x + a) perfectly divides p(x)
c)
If the degree of p(x) is less then the degree of d(x), we should not divide p(x) with d(x)
d)
Division of a polynomial with another polynomial stops when the degree of the remainder equals the degree of the divisor
(i)
{b,c,a}
(ii)
{a,c}
(iii)
{b,d,a}
(iv)
{b,a}
(v)
{d,c}
Question
40
40.
In which of the cases,
g
(
x
)
is a factor of
f
(
x
)
?
(i)
f
(
x
)
=
(
6
x
3
+
x
2
−
64
x
+
21
)
,
g
(
x
)
=
(
−
2
x
+
9
)
(ii)
f
(
x
)
=
(
2
x
3
−
9
x
2
−
18
x
+
81
)
,
g
(
x
)
=
(
2
x
+
7
)
(iii)
f
(
x
)
=
(
6
x
3
−
11
x
2
−
78
x
+
27
)
,
g
(
x
)
=
(
x
+
3
)
(iv)
f
(
x
)
=
(
4
x
3
−
16
x
2
−
51
x
+
189
)
,
g
(
x
)
=
(
x
+
2
)
(v)
f
(
x
)
=
(
−
2
x
3
−
5
x
2
+
19
x
+
42
)
,
g
(
x
)
=
(
−
3
x
+
1
)
Question
41
41.
If the polynomial
a
x
2
−
4
x
+
b
is divided by
(
x
−
4
)
,
it leaves a remainder
28
.
If it is divided by
(
x
−
1
)
,
it leaves a remainder
-5
.
Find the value of
a
and
b
(i)
3
,
-3
(ii)
-5
,
2
(iii)
3
,
-4
(iv)
-3
,
4
(v)
4
,
-4
Assignment Key
1) (iii)
2) (i)
3) (ii)
4) (ii)
5) (v)
6) (iii)
7) (iv)
8) (ii)
9) (v)
10) (ii)
11) (ii)
12) (i)
13) (v)
14) (i)
15) (ii)
16) (i)
17) (iv)
18) (i)
19) (i)
20) (i)
21) (iv)
22) (iii)
23) (ii)
24) (iii)
25) (v)
26) (v)
27) (i)
28) (i)
29) (iii)
30) (ii)
31) (i)
32) (i)
33) (iii)
34) (ii)
35) (i)
36) (i)
37) (i)
38) (iii)
39) (ii)
40) (iii)
41) (iii)
