Chapter 2: Polynomials

VVI questions — the ones the board asked most (20)

1. Q1Short answer2 marksAsked: 2020, 2022

   Find the zeroes of the polynomial 3x^2 - x - 4.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 65 s

   Model answer · zeroes of a quadratic polynomial

   1. 1.The given polynomial is 3x^2 - x - 4. By the method of splitting the middle term, we write -x as -4x + 3x, so that 3x^2 - x - 4 = 3x^2 - 4x + 3x - 4.1 mark
   2. 2.Taking common factors from the groups, 3x^2 - 4x + 3x - 4 = x(3x - 4) + 1(3x - 4) = (3x - 4)(x + 1). So, the value of 3x^2 - x - 4 is zero when 3x - 4 = 0 or x + 1 = 0, i.e., when x = 4/3 or x = -1. Therefore, the zeroes of 3x^2 - x - 4 are 4/3 and -1.1 mark

   • Write the splitting step explicitly as 3x^2 - 4x + 3x - 4 before factoring.
   • State the factor form (3x-4)(x+1) clearly before giving the zeroes.
   • End with both zeroes written together as 4/3 and -1.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

2. Q2Short answer2 marksAsked: 2025

   Divide x^3 + 1 by x + 1.

   

   Medha Ma’am

   AskMedha’s AI teacher

   Hear it from Medha Ma’am · 65 s

   Model answer · division of polynomials

   1. 1.Long division: write x^3 + 1 as x^3 + 0x^2 + 0x + 1. First term x^3 ÷ x = x^2; subtract x^2(x + 1) = x^3 + x^2 to leave −x^2 + 0x + 1. Next term −x^2 ÷ x = −x; subtract −x(x + 1) = −x^2 − x to leave x + 1. Next term x ÷ x = 1; subtract 1(x + 1) to leave 0.1 mark
   2. 2.So the quotient is x^2 − x + 1 and the remainder is 0, i.e. x^3 + 1 = (x + 1)(x^2 − x + 1).1 mark

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

3. Q3Short answer2 marksAsked: 2026

   Find the zeroes of the quadratic polynomial 3x^2 - x - 4.

   

   Medha Ma’am

   AskMedha’s AI teacher

   Hear it from Medha Ma’am · 60 s

   Model answer · zeroes of quadratic polynomial

   1. 1.We have p(x) = 3x^2 - x - 4. By the method of splitting the middle term, 3x^2 - x - 4 = 3x^2 - 4x + 3x - 4 = x(3x - 4) + 1(3x - 4)1 mark
   2. 2.So, 3x^2 - x - 4 = (3x - 4)(x + 1). Hence, the value of 3x^2 - x - 4 is zero when either 3x - 4 = 0 or x + 1 = 0, i.e., when x = 4/3 or x = -1. So, the zeroes of 3x^2 - x - 4 are 4/3 and -1.1 mark

   • Write the splitting step showing 3x^2 - 4x + 3x - 4 before factoring.
   • State the factors as (3x - 4)(x + 1) = 0 explicitly.
   • End with both zeroes written as x = 4/3 and x = -1.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

4. Q4Short answer2 marksAsked: 2026

   If x^2 + 2x - 3 is divided by x - 2 then find the remainder.

   How the model answer begins

   1. Given polynomial p(x) = x^2 + 2x - 3 is divided by x - 2. According to the Remainder Theorem, if a polynomial p(x) is divided by (x - a), the remainder is p(a). Here the divisor is (x - 2), so a = 2, and the remainder…

   The full answer from Medha Ma’am →
5. Q5Short answer2 marksAsked: 2026

   Find a quadratic polynomial of which the sum and product of zeros are - 7 and 10 respectively.

   How the model answer begins

   1. Let the quadratic polynomial be ax2 + bx + c, and its zeroes be α and β. If a = 1, the required polynomial is given by the formula x2 - (sum of zeroes)x + (product of zeroes), i.e., x2 - (α + β)x + αβ, where α + β =…

   The full answer from Medha Ma’am →
6. Q6Short answer2 marksAsked: 2026

   Divide y^6 + 3y^2 + 10 by y^3 + 1.

   How the model answer begins

   1. Dividend = y^6 + 3y^2 + 10, divisor = y^3 + 1. First term of the quotient = y^6 ÷ y^3 = y^3. y^3(y^3+1) = y^6 + y^3 (y^6 + 3y^2 + 10) − (y^6 + y^3) = 3y^2 − y^3 + 10

   The full answer from Medha Ma’am →
7. Q7Short answer2 marksAsked: 2022

   Find a quadratic polynomial p(x), the sum of whose zeroes are sqrt(2) and product is 1/3.

   How the model answer begins

   1. The standard form of a quadratic polynomial when the sum and product of its zeroes are given is p(x) = k[x² – (sum of zeroes)x + (product of zeroes)], where k is a constant.

   The full answer from Medha Ma’am →
8. Q8Short answer2 marksAsked: 2022

   Find a quadratic polynomial whose zeroes are sqrt(3)+1 and sqrt(3)-1.

   How the model answer begins

   1. The sum of the zeroes is (√3 + 1) + (√3 − 1) = 2√3.

   The full answer from Medha Ma’am →
9. Q9Short answer2 marksAsked: 2022

   Find the zeroes of the quadratic polynomial z^2 - 2z - 8.

   How the model answer begins

   1. Equating the quadratic polynomial z^2 - 2z - 8 to zero, we get z^2 - 2z - 8 = 0. By the method of splitting the middle term, z^2 - 2z - 8 = z^2 - 4z + 2z - 8 = z(z - 4) + 2(z - 4) = (z - 4)(z + 2). So the value of z^2 -…

   The full answer from Medha Ma’am →
10. Q10Short answer2 marksAsked: 2022

    Find a cubic polynomial whose zeroes are 3, - 1 and - 2.

    How the model answer begins

    1. Given: the zeroes are α = 3, β = −1 and γ = −2. So the cubic polynomial has the factors (x − 3)(x + 1)(x + 2). Sum of zeroes: α + β + γ = 3 + (−1) + (−2) = 0

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11. Q11Short answer2 marksAsked: 2022

    If alpha, beta are the zeroes of polynomial p(x) = 2x^2 - 11x + 15 then find the value of 1/alpha + 1/beta.

    How the model answer begins

    1. For the quadratic polynomial p(x) = 2x^2 - 11x + 15, comparing with ax^2 + bx + c gives a = 2, b = -11, c = 15. Since α and β are the zeroes of p(x), the sum of zeroes α + β = -b/a = -(-11)/2 = 11/2 and the product of…

    The full answer from Medha Ma’am →
12. Q12Short answer2 marksAsked: 2022

    If - 2 is a zero of the polynomial 3x^2 + 4x + 2k, then find the value of k.

    How the model answer begins

    1. Since –2 is a zero of the polynomial p(x) = 3x^2 + 4x + 2k, a real number k is said to be a zero of a polynomial p(x), if p(k) = 0. Substituting x = –2 in p(x), we get p(–2) = 0, that is, 3(–2)^2 + 4(–2) + 2k = 0.

    The full answer from Medha Ma’am →
13. Q13Short answer2 marksAsked: 2023

    Find the zeroes of 4t^2 - 4t + 1 and verify the relation between zeroes and coefficients.

    How the model answer begins

    1. The zeroes of the quadratic polynomial 4t^2 - 4t + 1 are found by factorisation. 4t^2 - 4t + 1 = (2t - 1)^2 So, the value of 4t^2 - 4t + 1 is zero when 2t - 1 = 0 t = 1/2 Therefore, the zeroes of 4t^2 - 4t + 1 are 1/2…

    The full answer from Medha Ma’am →
14. Q14Short answer2 marksAsked: 2023

    Find a quadratic polynomial, the sum and product of whose zeroes are 0 and - 4 respectively.

    How the model answer begins

    1. A quadratic polynomial, when the sum and product of its zeroes are given, is found using the formula x² – (sum of zeroes)x + (product of zeroes).

    The full answer from Medha Ma’am →
15. Q15Short answer2 marksAsked: 2023

    Find a quadratic polynomial, the sum and product of whose zeroes are 0 and - 4 respectively.

    How the model answer begins

    1. A quadratic polynomial with sum of zeroes (α + β) and product of zeroes αβ is given by the formula x² - (α + β)x + αβ. Here the sum of zeroes is given as 0 and the product of zeroes is given as -4.

    The full answer from Medha Ma’am →
16. Q16Short answer2 marksAsked: 2023

    Find the zeroes of 4t^2 - 4t + 1 and verify the relation between zeroes and coefficients.

    How the model answer begins

    1. The zeroes of 4t^2 - 4t + 1 are found by splitting the middle term. We write 4t^2 - 4t + 1 = 4t^2 - 2t - 2t + 1 = 2t(2t - 1) - 1(2t - 1) = (2t - 1)(2t - 1) = (2t - 1)^2. So, the value of 4t^2 - 4t + 1 is zero when 2t -…

    The full answer from Medha Ma’am →
17. Q17Short answer2 marksAsked: 2021

    Find a quadratic polynomial whose sum of zeros is sqrt(2) and product of zeros is 1/3.

    How the model answer begins

    1. A quadratic polynomial, the sum and product of whose zeroes are given, is found using the formula: quadratic polynomial = x² – (sum of zeroes)x + (product of zeroes).

    The full answer from Medha Ma’am →
18. Q18Short answer2 marksAsked: 2021

    If alpha and beta are the zeros of the quadratic polynomial 5x^2 - 7x + 1, then what will be the value of (1/alpha + 1/beta)?

    How the model answer begins

    1. For the quadratic polynomial 5x^2 - 7x + 1, comparing with ax^2 + bx + c, we get a = 5, b = -7, c = 1. If alpha and beta are the zeroes of this polynomial, then sum of zeroes = alpha + beta = -b/a = -(-7)/5 = 7/5, and…

    The full answer from Medha Ma’am →
19. Q19Short answer2 marksAsked: 2021

    Find the zeroes of the quadratic polynomial x^2 + 11x + 30.

    How the model answer begins

    1. Equating the quadratic polynomial to zero, we have x^2 + 11x + 30 = 0. By the method of splitting the middle term, 11x is split as 5x + 6x, since 5 × 6 = 30 and 5 + 6 = 11. So, x^2 + 11x + 30 = x^2 + 5x + 6x + 30 = 0.

    The full answer from Medha Ma’am →
20. Q20Short answer2 marksAsked: 2021

    If (x - a) is a factor of the polynomial (x^2 - ax + a - 1), then find the value of a.

    How the model answer begins

    1. According to the Factor Theorem, a real number k is a zero of the polynomial p(x) if p(k) = 0, and if (x - k) is a factor of p(x), then p(k) = 0. Here the given polynomial is p(x) = x^2 - ax + a - 1, and it is given…

    The full answer from Medha Ma’am →

1 more questions of this chapter, each with its model answer, in the AskMedha app.

Objective questions with the board’s key (10)

1. 1. Bihar Board 2026

   Factors of the polynomial x^2 - 4x + 3 is

   • (A) (x - 1)(x + 3)
   • (B) (x + 1)(x - 3)
   • (C) (x - 1)(x - 3) ✓
   • (D) (x + 1)(x + 3)

   Answer: (C)

2. 2. Bihar Board 2026

   Zeroes of the polynomial x^2 + 7x + 10 are

   • (A) - 2, - 5 ✓
   • (B) 2, 5
   • (C) - 2, 5
   • (D) 2, - 5

   Answer: (A)

3. 3. Bihar Board 2026

   Sum of the zeroes of the polynomial 6x^2 - 7x - 3 is

   • (A) -7/6
   • (B) 7/6 ✓
   • (C) -1/2
   • (D) 1/2

   Answer: (B)

4. 4. Bihar Board 2026

   Product of the zeroes of the polynomial 2x^2 + 3x + 1 is

   • (A) -3/2
   • (B) 3/2
   • (C) -1/2
   • (D) 1/2 ✓

   Answer: (D)

5. 5. Bihar Board 2026

   Polynomial having zeroes -3 and 2 is

   • (A) x^2 - x - 6
   • (B) x^2 + x - 6 ✓
   • (C) x^2 - x + 6
   • (D) x^2 + x + 6

   Answer: (B)

6. 6. Bihar Board 2026

   The zeroes of the polynomial x^2 - 4 are

   • (A) 1, 2
   • (B) 2, 2
   • (C) 2, -2 ✓
   • (D) sqrt(2), -sqrt(2)

   Answer: (C)

7. 7. Bihar Board 2026

   If sum of the zeroes of the polynomial x^2 - kx + 8 is 6 then the value of k is

   • (A) 8
   • (B) 6 ✓
   • (C) -6
   • (D) 3

   Answer: (B)

8. 8. Bihar Board 2026

   If alpha and beta are the zeroes of the polynomial 2x^2 + 3x - 4 then the value of 5(alpha.beta) is

   • (A) 10
   • (B) -10 ✓
   • (C) 15
   • (D) 20

   Answer: (B)

9. 9. Bihar Board 2026

   What are the zeroes of polynomial 4x^2 + 5sqrt(2)x - 3 ?

   • (A) -3sqrt(2), sqrt(2)
   • (B) -3sqrt(2)/2, sqrt(2)/4 ✓
   • (C) -3sqrt(2), sqrt(2)/2
   • (D) 3sqrt(2), sqrt(2)

   Answer: (B)

10. 10. Bihar Board 2026

    If the zeroes of a quadratic polynomial are 5 and -3 then the quadratic polynomial is

    • (A) x^2 - 2x + 15
    • (B) x^2 - 2x - 15 ✓
    • (C) x^2 + 2x + 15
    • (D) x^2 - 15x - 15

    Answer: (B)

Questions parents and students ask

How many marks does "Polynomials" carry in Bihar Board Class 10 Maths?

In the Bihar Board papers of 2021–2026, this chapter carried about 7 marks and about 12 questions a year; by marks it ranks 8 in Maths.

Which are the most important (VVI) questions of "Polynomials"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 21 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

BSEB does not publish a marking scheme. Each answer is written point by point on a structure derived from past papers and checked by a second, independent model; it is never called the board’s official scheme.