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NCERT Solutions
NCERT Solutions
Polynomials
30 Solutions
Exercise:
All Exercises
EXERCISE 2.1
EXERCISE 2.2
EXERCISE 2.3
EXERCISE 2.4
Q1
EXERCISE 2.1
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i)
4
x
2
−
3
x
+
7
4 x^{2}-3 x+7
4
x
2
−
3
x
+
7
(ii)
y
2
+
2
y^{2}+\sqrt{2}
y
2
+
2
(iii)
3
t
+
t
2
3 \sqrt{t}+t \sqrt{2}
3
t
+
t
2
(iv)
y
+
2
y
y+\frac{2}{y}
y
+
y
2
(v)
x
10
+
y
3
+
t
50
x^{10}+y^{3}+t^{50}
x
10
+
y
3
+
t
50
Q2
EXERCISE 2.1
Write the coefficients of
x
2
x^{2}
x
2
in each of the following:
(i)
2
+
x
2
+
x
2+x^{2}+x
2
+
x
2
+
x
(ii)
2
−
x
2
+
x
3
2-x^{2}+x^{3}
2
−
x
2
+
x
3
(iii)
π
2
x
2
+
x
\frac{\pi}{2} x^{2}+x
2
π
x
2
+
x
(iv)
2
x
−
1
\sqrt{2} x-1
2
x
−
1
Q3
EXERCISE 2.1
Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Q4
EXERCISE 2.1
Write the degree of each of the following polynomials:
(i)
5
x
3
+
4
x
2
+
7
x
5 x^{3}+4 x^{2}+7 x
5
x
3
+
4
x
2
+
7
x
(ii)
4
−
y
2
4-y^{2}
4
−
y
2
(iii)
5
t
−
7
5 t-\sqrt{7}
5
t
−
7
(iv)
3
Q5
EXERCISE 2.1
Classify the following as linear, quadratic and cubic polynomials:
(i)
x
2
+
x
x^{2}+x
x
2
+
x
(ii)
x
−
x
3
x-x^{3}
x
−
x
3
(iii)
y
+
y
2
+
4
y+y^{2}+4
y
+
y
2
+
4
(iv)
1
+
x
1+x
1
+
x
(v)
3
t
3 t
3
t
(vi)
r
2
r^{2}
r
2
(vii)
7
x
3
7 x^{3}
7
x
3
Q1
EXERCISE 2.2
Find the value of the polynomial
5
x
−
4
x
2
+
3
5 x-4 x^{2}+3
5
x
−
4
x
2
+
3
at
(i)
x
=
0
x=0
x
=
0
(ii)
x
=
−
1
x=-1
x
=
−
1
(iii)
x
=
2
x=2
x
=
2
Q2
EXERCISE 2.2
Find
p
(
0
)
,
p
(
1
)
p(0), p(1)
p
(
0
)
,
p
(
1
)
and
p
(
2
)
p(2)
p
(
2
)
for each of the following polynomials:
(i)
p
(
y
)
=
y
2
−
y
+
1
p(y)=y^{2}-y+1
p
(
y
)
=
y
2
−
y
+
1
(ii)
p
(
t
)
=
2
+
t
+
2
t
2
−
t
3
p(t)=2+t+2 t^{2}-t^{3}
p
(
t
)
=
2
+
t
+
2
t
2
−
t
3
(iii)
p
(
x
)
=
x
3
p(x)=x^{3}
p
(
x
)
=
x
3
(iv)
p
(
x
)
=
(
x
−
1
)
(
x
+
1
)
p(x)=(x-1)(x+1)
p
(
x
)
=
(
x
−
1
)
(
x
+
1
)
Q3
EXERCISE 2.2
Verify whether the following are zeroes of the polynomial, indicated against them.
(i)
p
(
x
)
=
3
x
+
1
,
x
=
−
1
3
p(x)=3 x+1, x=-\frac{1}{3}
p
(
x
)
=
3
x
+
1
,
x
=
−
3
1
(ii)
p
(
x
)
=
5
x
−
π
,
x
=
4
5
p(x)=5 x-\pi, x=\frac{4}{5}
p
(
x
)
=
5
x
−
π
,
x
=
5
4
(iii)
p
(
x
)
=
x
2
−
1
,
x
=
1
,
−
1
p(x)=x^{2}-1, x=1,-1
p
(
x
)
=
x
2
−
1
,
x
=
1
,
−
1
(iv)
p
(
x
)
=
(
x
+
1
)
(
x
−
2
)
,
x
=
−
1
,
2
p(x)=(x+1)(x-2), x=-1,2
p
(
x
)
=
(
x
+
1
)
(
x
−
2
)
,
x
=
−
1
,
2
(v)
p
(
x
)
=
x
2
,
x
=
0
p(x)=x^{2}, x=0
p
(
x
)
=
x
2
,
x
=
0
(vi)
p
(
x
)
=
l
x
+
m
,
x
=
−
m
l
p(x)=l x+m, x=-\frac{m}{l}
p
(
x
)
=
l
x
+
m
,
x
=
−
l
m
(vii)
p
(
x
)
=
3
x
2
−
1
,
x
=
−
1
3
,
2
3
p(x)=3 x^{2}-1, x=-\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}
p
(
x
)
=
3
x
2
−
1
,
x
=
−
3
1
,
3
2
(viii)
p
(
x
)
=
2
x
+
1
,
x
=
1
2
p(x)=2 x+1, x=\frac{1}{2}
p
(
x
)
=
2
x
+
1
,
x
=
2
1
Q4
EXERCISE 2.2
Find the zero of the polynomial in each of the following cases:
(i)
p
(
x
)
=
x
+
5
p(x)=x+5
p
(
x
)
=
x
+
5
(ii)
p
(
x
)
=
x
−
5
p(x)=x-5
p
(
x
)
=
x
−
5
(iii)
p
(
x
)
=
2
x
+
5
p(x)=2 x+5
p
(
x
)
=
2
x
+
5
(iv)
p
(
x
)
=
3
x
−
2
p(x)=3 x-2
p
(
x
)
=
3
x
−
2
(v)
p
(
x
)
=
3
x
p(x)=3 x
p
(
x
)
=
3
x
(vi)
p
(
x
)
=
a
x
,
a
≠
0
p(x)=a x, a \neq 0
p
(
x
)
=
a
x
,
a
=
0
(vii)
p
(
x
)
=
c
x
+
d
,
c
≠
0
,
c
,
d
p(x)=c x+d, c \neq 0, c, d
p
(
x
)
=
c
x
+
d
,
c
=
0
,
c
,
d
are real numbers.
Q1
EXERCISE 2.3
Determine which of the following polynomials has
(
x
+
1
)
(x+1)
(
x
+
1
)
a factor :
(i)
x
3
+
x
2
+
x
+
1
x^{3}+x^{2}+x+1
x
3
+
x
2
+
x
+
1
(ii)
x
4
+
x
3
+
x
2
+
x
+
1
x^{4}+x^{3}+x^{2}+x+1
x
4
+
x
3
+
x
2
+
x
+
1
(iii)
x
4
+
3
x
3
+
3
x
2
+
x
+
1
x^{4}+3 x^{3}+3 x^{2}+x+1
x
4
+
3
x
3
+
3
x
2
+
x
+
1
(iv)
x
3
−
x
2
−
(
2
+
2
)
x
+
2
x^{3}-x^{2}-(2+\sqrt{2}) x+\sqrt{2}
x
3
−
x
2
−
(
2
+
2
)
x
+
2
Q2
EXERCISE 2.3
Use the Factor Theorem to determine whether
g
(
x
)
g(x)
g
(
x
)
is a factor of
p
(
x
)
p(x)
p
(
x
)
in each of the following cases:
(i)
p
(
x
)
=
2
x
3
+
x
2
−
2
x
−
1
,
g
(
x
)
=
x
+
1
p(x)=2 x^{3}+x^{2}-2 x-1, g(x)=x+1
p
(
x
)
=
2
x
3
+
x
2
−
2
x
−
1
,
g
(
x
)
=
x
+
1
(ii)
p
(
x
)
=
x
3
+
3
x
2
+
3
x
+
1
,
g
(
x
)
=
x
+
2
p(x)=x^{3}+3 x^{2}+3 x+1, g(x)=x+2
p
(
x
)
=
x
3
+
3
x
2
+
3
x
+
1
,
g
(
x
)
=
x
+
2
(iii)
p
(
x
)
=
x
3
−
4
x
2
+
x
+
6
,
g
(
x
)
=
x
−
3
p(x)=x^{3}-4 x^{2}+x+6, g(x)=x-3
p
(
x
)
=
x
3
−
4
x
2
+
x
+
6
,
g
(
x
)
=
x
−
3
Q3
EXERCISE 2.3
Find the value of
k
k
k
, if
x
−
1
x-1
x
−
1
is a factor of
p
(
x
)
p(x)
p
(
x
)
in each of the following cases:
(i)
p
(
x
)
=
x
2
+
x
+
k
p(x)=x^{2}+x+k
p
(
x
)
=
x
2
+
x
+
k
(ii)
p
(
x
)
=
2
x
2
+
k
x
+
2
p(x)=2 x^{2}+k x+\sqrt{2}
p
(
x
)
=
2
x
2
+
k
x
+
2
(iii)
p
(
x
)
=
k
x
2
−
2
x
+
1
p(x)=k x^{2}-\sqrt{2} x+1
p
(
x
)
=
k
x
2
−
2
x
+
1
(iv)
p
(
x
)
=
k
x
2
−
3
x
+
k
p(x)=k x^{2}-3 x+k
p
(
x
)
=
k
x
2
−
3
x
+
k
Q4
EXERCISE 2.3
Factorise :
(i)
12
x
2
−
7
x
+
1
12 x^{2}-7 x+1
12
x
2
−
7
x
+
1
(ii)
2
x
2
+
7
x
+
3
2 x^{2}+7 x+3
2
x
2
+
7
x
+
3
(iii)
6
x
2
+
5
x
−
6
6 x^{2}+5 x-6
6
x
2
+
5
x
−
6
(iv)
3
x
2
−
x
−
4
3 x^{2}-x-4
3
x
2
−
x
−
4
Q5
EXERCISE 2.3
Factorise :
(i)
x
3
−
2
x
2
−
x
+
2
x^{3}-2 x^{2}-x+2
x
3
−
2
x
2
−
x
+
2
(ii)
x
3
−
3
x
2
−
9
x
−
5
x^{3}-3 x^{2}-9 x-5
x
3
−
3
x
2
−
9
x
−
5
(iii)
x
3
+
13
x
2
+
32
x
+
20
x^{3}+13 x^{2}+32 x+20
x
3
+
13
x
2
+
32
x
+
20
(iv)
2
y
3
+
y
2
−
2
y
−
1
2 y^{3}+y^{2}-2 y-1
2
y
3
+
y
2
−
2
y
−
1
Q1
EXERCISE 2.4
Use suitable identities to find the following products:
(i)
(
x
+
4
)
(
x
+
10
)
(x+4)(x+10)
(
x
+
4
)
(
x
+
10
)
(ii)
(
x
+
8
)
(
x
−
10
)
(x+8)(x-10)
(
x
+
8
)
(
x
−
10
)
(iii)
(
3
x
+
4
)
(
3
x
−
5
)
(3 x+4)(3 x-5)
(
3
x
+
4
)
(
3
x
−
5
)
(iv)
(
y
2
+
3
2
)
(
y
2
−
3
2
)
(y^{2}+\frac{3}{2})(y^{2}-\frac{3}{2})
(
y
2
+
2
3
)
(
y
2
−
2
3
)
(v)
(
3
−
2
x
)
(
3
+
2
x
)
(3-2 x)(3+2 x)
(
3
−
2
x
)
(
3
+
2
x
)
Q2
EXERCISE 2.4
Evaluate the following products without multiplying directly:
(i)
103
×
107
103 \times 107
103
×
107
(ii)
95
×
96
95 \times 96
95
×
96
(iii)
104
×
96
104 \times 96
104
×
96
Q3
EXERCISE 2.4
Factorise the following using appropriate identities:
(i)
9
x
2
+
6
x
y
+
y
2
9 x^{2}+6 x y+y^{2}
9
x
2
+
6
x
y
+
y
2
(ii)
4
y
2
−
4
y
+
1
4 y^{2}-4 y+1
4
y
2
−
4
y
+
1
(iii)
x
2
−
y
2
100
x^{2}-\frac{y^{2}}{100}
x
2
−
100
y
2
Q4
EXERCISE 2.4
Expand each of the following, using suitable identities:
(i)
(
x
+
2
y
+
4
z
)
2
(x+2 y+4 z)^{2}
(
x
+
2
y
+
4
z
)
2
(ii)
(
2
x
−
y
+
z
)
2
(2 x-y+z)^{2}
(
2
x
−
y
+
z
)
2
(iii)
(
−
2
x
+
3
y
+
2
z
)
2
(-2 x+3 y+2 z)^{2}
(
−
2
x
+
3
y
+
2
z
)
2
(iv)
(
3
a
−
7
b
−
c
)
2
(3 a-7 b-c)^{2}
(
3
a
−
7
b
−
c
)
2
(v)
(
−
2
x
+
5
y
−
3
z
)
2
(-2 x+5 y-3 z)^{2}
(
−
2
x
+
5
y
−
3
z
)
2
(vi)
[
1
4
a
−
1
2
b
+
1
]
2
[\frac{1}{4} a-\frac{1}{2} b+1]^{2}
[
4
1
a
−
2
1
b
+
1
]
2
Q5
EXERCISE 2.4
Factorise:
(i)
4
x
2
+
9
y
2
+
16
z
2
+
12
x
y
−
24
y
z
−
16
x
z
4 x^{2}+9 y^{2}+16 z^{2}+12 x y-24 y z-16 x z
4
x
2
+
9
y
2
+
16
z
2
+
12
x
y
−
24
yz
−
16
x
z
(ii)
2
x
2
+
y
2
+
8
z
2
−
2
2
x
y
+
4
2
y
z
−
8
x
z
2 x^{2}+y^{2}+8 z^{2}-2 \sqrt{2} x y+4 \sqrt{2} y z-8 x z
2
x
2
+
y
2
+
8
z
2
−
2
2
x
y
+
4
2
yz
−
8
x
z
Q6
EXERCISE 2.4
Write the following cubes in expanded form:
(i)
(
2
x
+
1
)
3
(2 x+1)^{3}
(
2
x
+
1
)
3
(ii)
(
2
a
−
3
b
)
3
(2 a-3 b)^{3}
(
2
a
−
3
b
)
3
(iii)
[
3
2
x
+
1
]
3
[\frac{3}{2} x+1]^{3}
[
2
3
x
+
1
]
3
(iv)
[
x
−
2
3
y
]
3
[x-\frac{2}{3} y]^{3}
[
x
−
3
2
y
]
3
Q7
EXERCISE 2.4
Evaluate the following using suitable identities:
(i)
(
99
)
3
(99)^{3}
(
99
)
3
(ii)
(
102
)
3
(102)^{3}
(
102
)
3
(iii)
(
998
)
3
(998)^{3}
(
998
)
3
Q8
EXERCISE 2.4
Factorise each of the following:
(i)
8
a
3
+
b
3
+
12
a
2
b
+
6
a
b
2
8 a^{3}+b^{3}+12 a^{2} b+6 a b^{2}
8
a
3
+
b
3
+
12
a
2
b
+
6
a
b
2
(ii)
8
a
3
−
b
3
−
12
a
2
b
+
6
a
b
2
8 a^{3}-b^{3}-12 a^{2} b+6 a b^{2}
8
a
3
−
b
3
−
12
a
2
b
+
6
a
b
2
(iii)
27
−
125
a
3
−
135
a
+
225
a
2
27-125 a^{3}-135 a+225 a^{2}
27
−
125
a
3
−
135
a
+
225
a
2
(iv)
64
a
3
−
27
b
3
−
144
a
2
b
+
108
a
b
2
64 a^{3}-27 b^{3}-144 a^{2} b+108 a b^{2}
64
a
3
−
27
b
3
−
144
a
2
b
+
108
a
b
2
(v)
27
p
3
−
1
216
−
9
2
p
2
+
1
4
p
27 p^{3}-\frac{1}{216}-\frac{9}{2} p^{2}+\frac{1}{4} p
27
p
3
−
216
1
−
2
9
p
2
+
4
1
p
Q9
EXERCISE 2.4
Verify : (i)
x
3
+
y
3
=
(
x
+
y
)
(
x
2
−
x
y
+
y
2
)
x^{3}+y^{3}=(x+y)(x^{2}-x y+y^{2})
x
3
+
y
3
=
(
x
+
y
)
(
x
2
−
x
y
+
y
2
)
(ii)
x
3
−
y
3
=
(
x
−
y
)
(
x
2
+
x
y
+
y
2
)
x^{3}-y^{3}=(x-y)(x^{2}+x y+y^{2})
x
3
−
y
3
=
(
x
−
y
)
(
x
2
+
x
y
+
y
2
)
Q10
EXERCISE 2.4
Factorise each of the following:
(i)
27
y
3
+
125
z
3
27 y^{3}+125 z^{3}
27
y
3
+
125
z
3
(ii)
64
m
3
−
343
n
3
64 m^{3}-343 n^{3}
64
m
3
−
343
n
3
[Hint : See Question 9.]
Q11
EXERCISE 2.4
Factorise:
27
x
3
+
y
3
+
z
3
−
9
x
y
z
27 x^{3}+y^{3}+z^{3}-9 x y z
27
x
3
+
y
3
+
z
3
−
9
x
yz
Q12
EXERCISE 2.4
Verify that
x
3
+
y
3
+
z
3
−
3
x
y
z
=
1
2
(
x
+
y
+
z
)
[
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
]
x^{3}+y^{3}+z^{3}-3 x y z=\frac{1}{2}(x+y+z)[(x-y)^{2}+(y-z)^{2}+(z-x)^{2}]
x
3
+
y
3
+
z
3
−
3
x
yz
=
2
1
(
x
+
y
+
z
)
[(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
]
Q13
EXERCISE 2.4
If
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
, show that
x
3
+
y
3
+
z
3
=
3
x
y
z
x^{3}+y^{3}+z^{3}=3 x y z
x
3
+
y
3
+
z
3
=
3
x
yz
.
Q14
EXERCISE 2.4
Without actually calculating the cubes, find the value of each of the following:
(i)
(
−
12
)
3
+
(
7
)
3
+
(
5
)
3
(-12)^{3}+(7)^{3}+(5)^{3}
(
−
12
)
3
+
(
7
)
3
+
(
5
)
3
(ii)
(
28
)
3
+
(
−
15
)
3
+
(
−
13
)
3
(28)^{3}+(-15)^{3}+(-13)^{3}
(
28
)
3
+
(
−
15
)
3
+
(
−
13
)
3
Q15
EXERCISE 2.4
Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: Area :
25
a
2
−
35
a
+
12
25 a^{2}-35 a+12
25
a
2
−
35
a
+
12
(i) Area :
35
y
2
+
13
y
−
12
35 y^{2}+13 y-12
35
y
2
+
13
y
−
12
(ii)
Q16
EXERCISE 2.4
What are the possible expressions for the dimensions of the cuboids whose volumes are given below? Volume :
3
x
2
−
12
x
3 x^{2}-12 x
3
x
2
−
12
x
(i) Volume :
12
k
y
2
+
8
k
y
−
20
k
12 k y^{2}+8 k y-20 k
12
k
y
2
+
8
k
y
−
20
k
(ii)
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