CBSE Class 9 Polynomials Sure Shot Questions

Class 9 Mathematics Study Guide: CBSE Class 9 Polynomials Sure Shot Questions

Access comprehensive study materials and useful resources for Chapter 02 Polynomials using the CBSE Class 9 Polynomials Sure Shot Questions. Designed to align with the 2026-27 CBSE academic guidelines, these advanced resources help Class 9 Mathematics students reinforce core concepts beyond standard textbooks.

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1. Factorize the following: 9x2 + 6x + 1 – 25y2.

2. Factorize the following: a2 + b2 + 2ab + 2bc + 2ca

3. Show that p(x) = x3 – 3x2 + 2x – 6 has only one real zero.

4. Find the value of a if x + 6 is a factor of x3 + 3x2 + 4x + a.

5. If polynomials ax3 + 3x2 – 3 and 2x3 – 5x + a leaves the same remainder when each is divided byx – 4, find the value of a..

6. The polynomial f(x)= x4 – 2x3 +3x2 – ax + b when divided by (x – 1) and (x + 1) leaves the remainders 5 and 19 respective ly. Find the values of a and b. Hence, find the remainder when

f(x) is divided by (x – 2).

7. If the polynomials 2x3 +ax2 + 3x – 5 and x3 + x2 – 2x + a leave the same remainder when divided by (x – 2), find the value of a. Also, find the remainder in each case.

8. If the polynomials az3 + 4z2 + 3z – 4 and z3 – 4z + a leave the same remainder when divided by z – 3, find the value of a.

9. The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by x + 1 leaves the remainder 19. Find the values of a. Also find the remainder when p(x) is divided by x + 2.

10. If both x – 2 and x – 1/ 2 are factors of px2 + 5x + r, show that p = r.

11. Without actual division, prove that 2x4 – 5x3 + 2x2 – x + 2 is divisible by x2 – 3x + 2.

12. Simplify (2x – 5y)3 – (2x + 5y)3.

13. Multiply x2 + 4y2 + z2 + 2xy + xz – 2yz by (– z + x – 2y).

14. If a, b, c are all non-zero and a + b + c = 0, prove that 

useful-resources-polynomials-cbse-class-9-polynomials

15. If a + b + c = 5 and ab + bc + ca = 10, then prove that a3 + b3 + c3 –3abc = – 25.

16. Without actual division, prove that 2x4 – 6x3 +3x2 +3x – 2 is exactly divisible by x2 – 3x + 2.

17. Without actual division, prove that x3 – 3x2 – 13x + 15 is exactly divisible by x2 + 2x – 3.

18. Find the values of a and b so that the polynomial x3 – 10x2 +ax + b is exactly divisible by (x – 1) as well as (x – 2).

19. Find the integral zeroes of the polynomial 2x3 + 5x2 – 5x – 2.

20. If (x – 3) and x -1/3 are both factors of ax2 + 5x + b, then show that a = b.
 
21. Find the values of a and b so that the polynomial x4 + ax3 – 7x2 +8x + b is exactly divisible by (x + 2) as well as (x + 3).
 
22. If x3 + ax2 + bx + 6 has (x – 2) as a factor and leaves a remainder 3 when divided by (x – 3), find the values of a and b.
 
23. Find the value of x3 + y3 + 15xy – 125 if x + y = 5.
 
24. Without actually calculating, find the value of (25)3 – (75)3 + (50)3.
 
25. Factorise each of the following cubic expressions:
 
(i) 8x3 – y3 – 12x2y + 6xy2
 
(ii) 27q3 – 125p3 – 135q2p + 225qp2
 
(iii) 8x3 + 729 + 108x2 + 486x
 
(iv) 27x3 - 1/216-9/2x2 + 1/4x
 
26. Factorise:
 
(i) x3 + 216y3 + 8z3 – 36xyz
 
(ii) a3 – 64b3 – 27c3 – 36abc
 
28. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
 
29. Find a zero of the polynomial p(x) = 2x + 1.
 
30. Verify whether 2 and 0 are zeroes of the polynomial x2 – 2x.
 
31. Find the zero of the polynomial in each of the following cases:
 
(i) p(x) = x + 5 (ii) p(x) = x – 5 (iii) p(x) = 2x + 5
 
(iv) p(x) = 3x – 2 (v) p(x) = 3x (vi) p(x) = ax, a ≠ 0
 
32. Find the value of each of the following polynomials at the indicated value of variables:
 
(i) p(x) = 5x2 – 3x + 7 at x = 1.
 
(ii) q(y) = 3y3 – 4y + 11 at y = 2.
 
(iii) p(t) = 4t4 + 5t3 – t2 + 6 at t = a.
 
33. Divide p(x) by g(x), where p(x) = x + 3x2 – 1 and g(x) = 1 + x.
 
34. Divide the polynomial 3x4 – 4x3 – 3x –1 by x – 1.
 
35. Find the remainder obtained on dividing p(x) = x3 + 1 by x + 1.
 
36. Find the remainder when x4 + x3 – 2x2 + x + 1 is divided by x – 1.
 
37. Check whether the polynomial q(t) = 4t3 + 4t2 – t – 1 is a multiple of 2t + 1.
 
38. Check whether p(x) is a multiple of g(x) or not, where p(x) = x3 – x + 1, g(x) = 2 – 3x.
 
39. Check whether g(x) is a factor of p(x) or not, where p(x) = 8x3 – 6x2 – 4x + 3, g(x) = x/3-1/4
 
40. Find the remainder when x3 – ax2 + 6x – a is divided by x – a.
 
41. Examine whether x + 2 is a factor of x3 + 3x2 + 5x + 6 and of 2x + 4.
 
42. Find the value of k, if x – 1 is a factor of 4x3 + 3x2 – 4x + k.
 
43. Find the value of a, if x – a is a factor of x3– ax2 + 2x + a – 1.
 
44. Factorise 6x2 + 17x + 5
 
45. Factorise y2 – 5y + 6
 
46. Factorise x3 – 23x2 + 142x – 120.
 
47. Factorise :
(i) x3– 2x2 – x + 2 (ii) x3 – 3x2 – 9x – 5
 
(iii) x3+ 13x2 + 32x + 20 (iv) 2y3 + y2 – 2y – 1
 
48. Factorise : 4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz
 
49. Expand (4a – 2b – 3c)2.
 
50. Factorise 4x2 + y2 + z2 – 4xy – 2yz + 4xz.
 
51. If x + 1 is a factor of ax3 + x2 – 2x + 4a – 9, find the value of a.
 
52. By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial : x4 + 1; x –1
 
53. Find the zeroes of the polynomial : p(x) = (x – 2)2 – (x + 2)2
 
54. Factorise :
(i) x2 + 9x + 18 (ii) 6x2 + 7x – 3
 
(iii) 2x2 – 7x – 15 (iv) 84 – 2r – 2r2
 
55. Factorise :
 
(i) 2x3 – 3x2 – 17x + 30 (ii) x3 – 6x2 + 11x – 6
 
(iii) x3 + x2 – 4x – 4 (iv) 3x3 – x2 – 3x + 1
 
56. Using suitable identity, evaluate the following:
 
(i) 1033 (ii) 101 × 102 (iii) 9992
 
57. Factorise the following:
 
(i) 4x2 + 20x + 25
 
(ii) 9y2 – 66yz + 121z2
 
(iii) (2x+1/3)2 - (x-1/2)2
 
58. Factorise the following :
 
(i) 9x2 – 12x + 3 (ii) 9x2 – 12x + 4
 
59. If a + b + c = 9 and ab + bc + ca = 26, find a2 + b2 + c2.
 
60. Expand the following :
 
(i) (4a – b + 2c)2
 
(ii) (3a – 5b – c)2
 
(iii) (– x + 2y – 3z)2
 
61. Find the value of
 
(i) x3 + y3 – 12xy + 64, when x + y = – 4
 
(ii) x3 – 8y3 – 36xy – 216, when x = 2y + 6
 
62. Factorise the following :
 
(i) 9x2 + 4y2 + 16z2 + 12xy – 16yz – 24xz
 
(ii) 25x2 + 16y2 + 4z2 – 40xy + 16yz – 20xz
 
(iii) 16x2 + 4y2 + 9z2 – 16xy – 12yz + 24 xz
CBSE Class 9 Polynomials Sure Shot Questions
 
66. Without finding the cubes, factorise (x – 2y)3 + (2y – 3z)3 + (3z – x)3
 
67. Give possible expressions for the length and breadth of the rectangle whose area is given by 4a2 + 4a –3.
 
68. Factorise: (i) 1+ 64x3 (ii) a3 + 2 2b3
 
69. Evaluate each of the following using suitable identities:
(i) (104)3         (ii) (999)3
 
70. Factorise : 8x3 + 27y3 + 36x2y + 54xy2
 
71. Factorise : 8x3 + y3 + 27z3 – 18xyz
 
72. Verify : (i) x3 + y3 = (x + y) (x2 – xy + y2) (ii) x3 – y3 = (x – y) (x2 + xy + y2)
 
73. Factorise each of the following:
 
(i) 27y3 + 125z3 (ii) 64m3 – 343n3
 
74. Factorise : 27x3 + y3 + z3 – 9xyz
 
75. Without actually calculating the cubes, find the value of each of the following:
 
(i) (–12)3 + (7)3 + (5)3
 
(ii) (28)3 + (–15)3 + (–13)3
 
76. Find the following product :(2x – y + 3z) (4x2 + y2 + 9z2 + 2xy + 3yz – 6xz) 77. Factorise :
 
(i) a3 – 8b3 – 64c3 – 24abc (ii) 2 2 a3 + 8b3 – 27c3 + 18 2 abc.
 
78. Give possible expressions for the length and breadth of rectangles, in which its areas is given by 35y2 + 13y –12
 
79. Without actually calculating the cubes, find the value of :
 
(i) (1/2)3 +(1/3)3 - (5/6)
 
(ii) (0.2)3 - (0.3)3 + (0.1)3
 
80. By Remainder Theorem find the remainder, when p(x) is divided by g(x), where
(i) p(x) = x3 – 2x2 – 4x – 1, g(x) = x + 1
 
(ii) p(x) = x3 – 3x2 + 4x + 50, g(x) = x – 3
 
(iii) p(x) = 4x3 – 12x2 + 14x – 3, g(x) = 2x – 1
 
(iv) p(x) = x3 – 6x2 + 2x – 4, g(x) = 1 3/2x
 
81. Check whether p(x) is a multiple of g(x) or not :
 
(i) p(x) = x3 – 5x2 + 4x – 3, g(x) = x – 2
 
(ii) p(x) = 2x3 – 11x2 – 4x + 5, g(x) = 2x + 1
 
82. Show that p – 1 is a factor of p10 – 1 and also of p11 – 1.
 
83. For what value of m is x3 – 2mx2+ 16 divisible by x + 2 ?
 
84. If x + 2a is a factor of x5 – 4a2x3 + 2x + 2a + 3, find a.
 
85. Find the value of m so that 2x – 1 be a factor of 8x4 + 4x3 – 16x2 + 10x + m.
 
86. Show that :
 
(i) x + 3 is a factor of 69 + 11x – x2 + x3 .
 
(ii) 2x – 3 is a factor of x + 2x3 – 9x2 + 12 .
 
87. If x + y = 12 and xy = 27, find the value of x3 + y3
 
88. Without actually calculating the cubes, find the value of 483 – 303 – 183.
 
89. Without finding the cubes, factorise (2x – 5y)3 + (5y – 3z)3 + (3z – 2x)3.
 
90. Without finding the cubes, factorise (x – y)3 + (y – z)3 + (z – x)3.
 

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