POLYNOMIAL’S
CLASS IX
Questions From CBSE Examination Papers
1.
2.
If a polynomial f (x) is divided by x – a, then
remainder is :
3.
(b) f (a)
(c) f (–a)
(d) f (a) – f(0)
(b) 3
(c) –2
x3
One of the factors of (
(d) 1
(c) x – 1
(b) –10
(d) x + 4
(c) –3
(d) 17
One of the factors of ( x – 1) –
– 1) is :
(a) x2 – 1 (b) x + 1
(c) x – 1
(d) x + 4
6.
b)3
8.
The factors of (2 a –
+ (b –
is :
(a) (2 a – b)(b – 2c)(c – a)
(b) 3(2a – b)(b – 2c)(c – a)
( c)
6(2a – b)(b – 2c)(c – a)
(d)
2a × b × 2 c
Which of the following is a binomial in y ?
1
+
2
(a) y2 +
2
(b) y +
y
17.
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18.
(c) x x +
x+
1
(d) x3 + 2x
a2 + b2 + c2 – ab – bc – ca equals :
(c) ( a – b + c)2
(b) 31
(c) 21
(d)
(d) 22
z1/3
If
+
+
= 0, then which one of the
following expressions is correct ?
19.
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20.
(c) x + y + z = 3xyz
( x + 2) is a factor of 2x3 + 5x2 – x – k. The value
of k is :
(a) 6
(b) –24
(c) –6
15
6
(b)
17
6
If x51 + 51 is divided by (x + 1), the remainder
is :
(a) 0
(b) 1
(c) 49
(d) 50
2 is a polynomial of degree :
(c)
1
6
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(d)
13
6
(b) 0
(c) 1
(d)
1
2
Which of the following is a polynomial in one
variable ?
(a) 3 – x2 + x
(d) 24
What is the remainder when x3 – 2x2 + x + 1 is
divided by (x – 1)?
(a) 0
(b) –1
(c) 1
(d) 2
2
x
x
2+
+
x2 , then p(–1) is :
If p ( x) =
2
3
(a)
21.
(b) (a – b – c)2
1
[( a - b)2 +
(b - c) 2 +
(c - a)2 ]
2
(a) 2
(d) x3 + y3 + z3 = 3xyz
12.
(b) x3 + x2 + x
(b) –2x3 + x2 – x – 19
(b) x + y + z = 3x1/3y1/3z1/3
11.
(a) x3 + 1
(c) 4 x3 + 13x – 25
If P(x) = 7 – 3x + 2x2, then value of P(–2) is :
1
(d) y y +
Which of the following is a trinomial in x ?
(a) ( a + b + c)2
(a) x3 + y3 + z3 = 0
10.
y+
2y
(c)
+ 8(c – a)
(b) x3 + x2 + x + 4
y1/3
3x+
1
16.
(a) 4 x3 – 13x + 6
x1/3
(d)
The remainder when x2 + 2 x + 1 is divided by
(x + 1) is :
(a) 4
(b) 0
(c) 1
(d) –2
3
In which of the following ( x + 2) is a factor ?
(a) 12
9.
2c)3
Which of the following is a polynomial inx ?
1
(a) x +
(b) x2 +
x
x
15.
(x2
5.
Zero of the polynomialp(x) where p(x) = ax, a ≠ 0
is :
1
(a) 1
(b) a
(c) 0
(d)
a
(c) x ++
2 x2 1
– 1) – (x – 1) is :
The coeffcient of x2 in (2 – 3x2)(x2 – 5) is :
(a) –17
7.
14.
The coefficient of x in the product of
(x – 1)(1 – 2x) is :
(a) x + 1 (b) x2 – 1
4.
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(a) f (0)
(a) –3
13.
(b)
3x+
4
1
(d) x +
x
The value of p for which x + p is a factor of
x2 + px + 3 – p is :
(c) x3 + y3 + 7
22.
(a) 1
23.
(b) –1
(c) 3
(d) –3
The degree of the polynomial p(x) = 3 is :
(a) 3
(b) 1
(c) 0
(d) 2
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24.
If
x y
+
=
- 1,( x, y ≠0), the value of x3 – y3 is :
y x
(a) 1
25.
26.
27.
(b) –1
32.
29.
30.
(b) 5
(c) –8
(d) 8
One of the factors of (16 y2 – 1) + (1 – 4y)2 is :
(a) (4 + y) (b) (4 – y) (c) (4 y + 1) (d) 8 y
If x2 + kx + 6 = (x + 2)(x + 3) for all x, the value
of k is :
(a) 1
(b) –1
(c) 5
(d) 3
Zero of the zero polynomial is :
(a) 0
(b) 1
(c) any real number
(d) not defned
If ( x – 1) is a factor of p(x) = x2 + x + k, then
value of k is :
(b) 2
(c) –2
(d) 1
Evaluate using suitable identity (999) 3.
x2
– x – 4.
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Factorise : 3
33.
Using factor theorem, show that ( x + 1) is a factor
of x19 + 1.
35.
36.
37.
38.
39.
Check whether the polynomial
p(s)=3s3+s2 -20s +12 is a multiple of 3 s – 2.
49.
Factorise : 125 x3 + 27y3.
50.
Find the value of x3 + y3 – 12xy + 64 when
x + y = –4.
51.
If x = 2y + 6, then fnd the value of x3 – 8y3
– 36xy – 216.
52.
Factorise : 27( x + y)3 – 8(x – y)3.
53.
Factorise : ( x – 2y)3 + (2y – 3z)3 + (3z – x)3.
54.
If 2 a =3+2b , prove that 8a3 –8 b3 – 36ab = 27
55.
If a – b = 7, a2 + b2 = 85, fnd a3 – b3.
56.
The polynomials kx3 + 3x2 – 8 and 3x3 – 5x + k
are divided by x + 2. If the remainder in each case
is the same, fnd the value of k.
57.
Find the values of a and b so that the polynomial
x3 + 10x2 + ax + b has (x – 1) and (x + 2) as
factors.
58.
Factorise : 8 x3 + y3 + 27z3 – 18xyz.
59.
If a2 + b2 + c2 = 90 and a + b + c = 20, then
fnd the value of ab + bc + ca.
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32.
34.
48.
The coeffcient of x2 in (3x2 – 5)(4 + 4x2) is :
(a) 3
31.
If – 1 is a zero of the polynomial
p(x) = ax3 - x2+ x+4 , find the : value of a
(d) 0
(1 + 3 x)3 is an example of :
(a) monomial
(b) binomial
(c) trinomial
(d) none of these
Degree of zero polynomial is :
(a) 0
(b) 1
(c) any natural number (d) not defned
(a) 12
28.
(c) 1/2
47.
Without actually calculating the cubes, fnd the
value of 303 + 203 – 503.
Evaluae (104)
Find
the
3
60.
61.
Factorise : 64 a3 – 27b3 – 144a2b + 108ab2.
62.
Simplify : ( a + b + c ) 2 +(
+ (a + b – c)2.
using suitable identity.
value
of
the
polynomial
17 when z = 3.
p( z ) =
3z 2 - 4 z +
Check whether the polynomial t + 1 is a factor of
4t3 + 4t2 – t – 1.
x 1
Factorise : x2 +
- .
4 8
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1
9
1
Factorise : 27p3 - p2 +
p.
216 2
4
-1
If x=
is a zero of the polynomial p(x) = 27x3
3
– ax2 – x + 3, then fnd the value of a.
63.
64.
a – b + c )2
Factorise : 4( x2 + 1)2 + 13(x2 + 1) – 12.
2
1
+
2 - 2x - .
Factorise : x2 +
2
x
x
65.
Determine whether (3 x – 2) is a factor of
3x3 + x2 – 20x + 12 ?
66.
Factorise:(2 x – y – z)3 + (2y – z – x)3 + (2z – x – y)3.
67.
If a + b = 11, a2 + b2 = 61, fnd a3 + b3.
40.
If 2 x + 3y = 8 and xy = 4, then fnd the value of
4x2 + 9y2.
68.
30 and a +
a2 +
b2 +
c2 =
b+
c=
10, then fnd the
value of ab +
bc +
ca.
41.
Check whether the polynomial 3 x – 1 is a factor
of 9x3 – 3x2 + 3x – 1.
Using factor theorem, show that (2 x + 1) is a factor
of 2x3 + 3x2 – 11x – 6.
69.
Using suitable identity evaluate :
(42)3 - (18)3 - (24)3 .
70.
Find the values of p and q, if the polynomial
x4 +px 3 + 2x2 – 3x + q is divisible by the polynomial
x2 – 1.
71.
Simplify ( x + y + z)2 – (x + y – z)2.
72.
Factorise 9x2 + y2 + z2 – 6xy + 2yz – 6zx. Hence
fnd its value if x = 1, y = 2 and z = –1.
42.
43.
Check whether (x +1) isa factor of x3+ x+ x2 +1
44.
Find the value of
a if (x – 1) is a factor of
2 x2 +
ax +
2.
45.
Factorise : 7 2x2 - 10 x - 4 2 .
46.
If a + b + c = 7 and ab + bc + ca = 20, fnd the
value of a2 + b2 + c2.
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73.
Find the value of a3 + b3 + 6ab – 8 when a + b =2
74.
If x + y + z = 9, then fnd the value of (3 – x)3
+ (3 – y)3 + (3 – z )3 – 3(3 – x)(3 – y)(3 – z).
75.
1
x3 +
y3 +
z3 - 3 xy =
(x+
y+
z)
2
[( x - y)2 +
( y - z) 2 +
( z - x)2 ].
95.
Verify :
If x – 3 is a factor of x2 – kx + 12, then fnd the
value of k. Also, fnd the other factor for this value
of k.
96.
2
2 3
(b2 - c2 ) 3 +
(c2 - a2 )3
Simplify : (a - b ) +
.
(a - b)3 +
(b - c) 3 +
(c - a)3
76.
Find the value of x3 + y3 + 9xy – 27, if x + y =3
97.
Prove that : 2 x3 + 2y3 + 2z3 – 6xyz = (x + y + z)
77.
If a + b + c = 6, then find the value of (2 – a)3
+ (2 – b)3 + (2 – c)3 – 3(2 – a)(2 – b)(2 – c).
78.
79.
80.
81.
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Factorise : 2 x3 – x2 – 13x – 6.
83.
Factorise : a3(b – c)3 + b3(c – a)3 + c3(a – b)3.
84.
If p = 4 – q, prove that p3 + q3 + 12pq = 64.
85.
Find the value of k so that 2x – 1 be a factor of
8x4 + 4x3 – 16x2 + 10x + k.
86.
x4
Using factor theorem show that x2 + 5x + 6 is
factor of x4 + 5x3 + 9x2 + 15x + 18.
99.
Prove that
(x +
y+
z) ×
[( x- y)2 +
( y - z) 2 ] =
2( x3 +
y3 +
z3 - 3xyz)
100.
The polynomials p(x) = ax3 + 4x2 + 3x – 4 and
q(x) = x3 – 4x + a leave the same remainder when
divided by x – 3. Find the remainder when p(x) is
divided by (x – 2).
101.
If both ( x + 2) and (2x + 1) are factors of
ax2+ 2 x + b, prove that a – b = 0.
102.
Simplify by factorisation method :
6 - 2 2x - x2
.
2 - x2
Show that ( x – 1) is a factor of P(x) = 3x3 – x2
– 3x + 1 and hence factorise P(x).
103.
ax + 8
x3
104.
The polynomials x3 + 2x2 – 5ax – 8 and x3 + ax2
– 12x – 6 when divided by (x – 2) and (x – 3)
leave remaindens p and q respectively. If q – p = 10
find the value of a .
x2
105.
Prove that ( x + y )3 –( x – y )3 –6 y(x2 – y2) = 8y3.
106.
Find the value of ( x – a)3 + (x – b)3 + (x – c)3
– 3(x – a)(x – b)(x – c), if a + b + c = 3x.
Simplify by factorisation method :
87.
If the polynomial P(x) =
–2
+3
–
is divided by (x – 2), it leaves a remainder 10.
Find the value of a
88.
Simplify : ( a + b + c)2 – ( a – b – c)2.
107.
89.
Factorise ( x – 3y)3 + (3y – 7z)3 + (7z – x)3.
108.
90.
Factorise : 2 2a3 +
8b3 - 27c3 +
18 2abc.
91.
Factorise : x6 – y6.
92.
Find the value of a if (x + a) is a factor of
x4 – a2x2 + 3x – a.
93.
Factorise by splitting the middle term :
9(x –
94.
2y)2
– 4(x – 2y) – 13.
9 - 2 3x - x2
3 - x2
If p(x) = x3 – ax2 + bx + 3 leaves a remainder
–19 when divided by (x + 2) and a remainder 17
when divided by (x – 2), prove that a + b = 6.
109.
The volume of a cube is given by the polynomial
p(x) = x3 –6 x2 + 12x – 8. Find the possible
expressions for the sides of the cube. Verify the
truth of your answer when the length of cube is
3 cm.
110.
Using factor theorem, factorise the polynomial :
x4 + 3x3 + 2x2 – 3x – 3.
111.
Factorise a7 + ab6.
112.
Using factor theorem, factorise the polynomial.
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Find the remainder obtained on dividing
1
2x4 - 3x3 - 5x2 +
x+
1 by x- .
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2
98.
What are the possible expressions for the dimensions
of the cuboids whose volume is given below ?
Volume = 12 ky2 + 8ky – 20k.
2
2(7)3 + 2(9)3 + 2(13)3 – 6(7) (9) (13).
If a2 + b2 + c2 = 250 and ab + bc + ca = 3,
find a + b + c.
1
1
3
.
If x +
=
7, then find the value of x +
x
x3
1
1
3
If x - =
3, then fnd the value of x - 3 .
x
x
3
2
If ax + bx + x – 6 has (x + 2) as a factor and
leaves a remainder 4 when divided by x – 2, fnd
the values of a and b.
82.
2
[( x – y) + (y – z) + (z – x) ]. Hence evaluate
x4 + 2x3 – 7 x2 – 8x + 12.
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4
3
2
113.
Without actual division, show that the polynomial
2x4 – 5x3 + 2x2 – x + 2 is exactly divisible by
x2 – 3x + 2.
125.
Without actual division, prove that 2 x – 8x. + 3x
+ 12x – 9 is exactly divisible by x2 – 4x + 3.
114.
If x and y be two positive real numbers such that
8x3 + 27y3 = 730 and 2x2y +3 xy2 = 15, then
evaluate 2x + 3y.
126.
If f(x) = x4 – 2x3 + 3x2 – ax + b is divided by
(x – 1) and (x + 1), it leaves the remainders 5 and
19 respectively. Find a and b.
115.
Factorise : ( x2 – 2x)2 – 2(x2 – 2x) – 3.
127.
If x2 – 3x + 2 is a factor of x4 – ax2 + b then fnd
a and b.
128.
Without actual division show that x4 + 2x3 – 2x2
+ 2x – 3 is exactly divisible by x2 + 2x – 3.
129.
1
27a2
9a
Factorise : 27a3 +
+
+
.
3
4
b
16b2
64b
116.
1
=
51, fnd
If x2 +
x2
1
1
x(i)
(ii) x3 - 3 .
x
x
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117.
Find the values of m and n so that the polynomial
f(x) = x3 – 6x2 + mx – n is exactly divisible by
(x – 1) as well as (x – 2).
118.
Factorise : x8 – y8.
119.
x4
2x3
Without actual division prove that
+
–
+ 2x – 3 is exactly divisible by x2 + 2x – 3.
130.
Find the values of a and b so that (x + 1) and
(x – 2) are factors of (x3 + ax2 + 2x + b).
131.
Wi t h o u t a c t u a l d i v i s i o n , p r o v e t h a t
(2x4 - 6x3 +
3x2 +
3x - 2) is exactly divisible by
2x2
( x2 - 3 x +
2).
120.
Factorise : a12x4 – a4x12.
132.
Simplify : (5 a + 3b)3 – (5a – 3b)3.
121.
Without actual division, prove that the polynomial
2x4 – 5x3 + 2x2 – x + 2 is exactly divisible by
x2 – 3x + 2.
133.
Find the value of a if (x – a) is a factor of x5
– a2x3 + 2 x + a + 3, hence factorisex2 – 2 ax – 3.
122.
Factorise : ( x2 – 3x)2 – 8(x2 – 3x) – 20.
134.
The polynomial ax3 + 3x2 – 3 and 2x3 – 5x + a
when divided by x – 4 leave the same remainder
in each case. Find the value of a.
135.
Factorise : 3 u3 – 4u2 – 12u + 16.
1
1
.
If x +
=
5 , then evaluate x6 +
x
x6
Multiply 9 x2 + 25y2 + 15xy + 12x – 20y + 16 by
3x – 5y – 4 using suitable identity.
123.
124.
The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a
– 7 when divided by (x + 1), leaves the remainder
19. Find the value of a. Also, fnd the remainder,
when p(x) is divided by x + 2.
Find the values of a and b so that (x + 1) and
(x – 1) are factors of x4 + ax3 – 3x2 + 2x + b.
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136.
137.
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