Class 10 Maths · Chapter 2

Polynomials

58 Class 10 Maths questions on Polynomials from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 1 of 3

1. Q1 NCERT · Exercise 2.1
   The graphs of $y=p(x)$ are given in Fig. 2.10 below, for ​some polynomials $p(x)$. ​Find the number of ​zeroes of $p(x)$, in each case. (i) Find ​the number of zeroes of $p(x)$ ​from graph

   1. (i).
   2. (ii)Find the number of zeroes of $p(x)$ from graph (ii). (iii) Find ​the number of zeroes of $p(x)$ from graph (iii). (iv) Find the number of zeroes of $p(x)$ from ​graph (iv). (v) Find the number of zeroes of $p(x)$ from graph (v). (vi) Find the number of zeroes ​of $p(x)$ from graph (vi).

   Show answer

   (i) $0$; (ii) $1$; (iii) $3$; (iv) $2$; (v) $4$; (vi) $3$.

   Step-by-step solution →
2. Q2 NCERT · Exercise 2.2
   Find ​the ​zeroes of ​the following ​quadratic polynomials and verify the relationship between ​the zeroes ​and the coefficients. ​

   1. (i)$x^2-2x-8$
   2. (ii)$4s^2-4s+1$
   3. (iii)$6x^2-3-7x$ ​
   4. (iv)$4u^2+8u$
   5. (v)$t^2-15$
   6. (vi)$3x^2-x-4$
   Show answer

   (i) $4,-2$; (ii) $\frac{1}{2}$ (repeated); (iii) $-\frac{1}{3},\frac{3}{2}$; (iv) $0,-2$; (v) $\sqrt{15},-\sqrt{15}$; (vi) $\frac{4}{3},-1$. In every case the sum of the zeroes equals $-\frac{b}{a}$ and the product equals $\frac{c}{a}$.

   Step-by-step solution →
3. Q3 NCERT · Exercise 2.2
   Find a quadratic ​polynomial each with ​the ​given numbers as the sum and ​product of its ​zeroes ​respectively.

   1. (i)$\frac{1}{4},-1$
   2. (ii)$\sqrt{2},\frac{1}{3}$
   3. (iii)$0,\sqrt{5}$
   4. (iv)$1,1$ ​
   5. (v)$-\frac{1}{4},\frac{1}{4}$
   6. (vi)$4,1$
   Show answer

   (i) $4x^2-x-4$; (ii) $3x^2-3\sqrt{2}x+1$; (iii) $x^2+\sqrt{5}$; (iv) $x^2-x+1$; (v) $4x^2+x+1$; (vi) $x^2-4x+1$.

   Step-by-step solution →
4. Q4 Exemplar · Exercise 2.1
   If ​one ​of ​the zeroes ​of the ​quadratic ​polynomial $(k-1)x^2+kx+1$ ​is $-3$, then the value of $k$ ​is

   1. (a)$\frac{4}{3}$
   2. (b)$\frac{-4}{3}$
   3. (c)$\frac{2}{3}$
   4. (d)$\frac{-2}{3}$
   Show answer

   (a) $\frac{4}{3}$

   Step-by-step solution →
5. Q5 Exemplar · Exercise 2.1
   A quadratic polynomial, ​whose ​zeroes ​are $-3$ ​and $4$, ​is

   1. (a)$x^2-x+12$
   2. (b)$x^2+x+12$
   3. (c)$\frac{x^2}{2}-\frac{x}{2}-6$
   4. (d)$2x^2+2x-24$
   Show answer

   (c) $\frac{x^2}{2}-\frac{x}{2}-6$

   Step-by-step solution →
6. Q6 Exemplar · Exercise 2.1
   If ​the zeroes ​of ​the quadratic ​polynomial $x^2+(a+1)x+b$ ​are 2 ​and $-3$, then

   1. (a)$a=-7,b=-1$
   2. (b)$a=5,b=-1$
   3. (c)$a=2,b=-6$
   4. (d)$a=0,b=-6$
   Show answer

   (d) $a=0,b=-6$

   Step-by-step solution →
7. Q7 Exemplar · Exercise 2.1
   The ​number ​of ​polynomials having ​zeroes as $-2$ and $5$ ​is

   1. (a)$1$
   2. (b)$2$
   3. (c)$3$
   4. (d)more than $3$
   Show answer

   (d) more than $3$

   Step-by-step solution →
8. Q8 Exemplar · Exercise 2.1
   Given that one of the ​zeroes of ​the cubic polynomial $a{x}^3+b{x}^2+cx+d$ is ​zero, the product of ​the ​other two ​zeroes ​is

   1. (a)$-\frac{c}{a}$
   2. (b)$\frac{c}{a}$
   3. (c)$0$
   4. (d)$-\frac{b}{a}$
   Show answer

   (b) $\frac{c}{a}$

   Step-by-step solution →
9. Q9 Exemplar · Exercise 2.1
   If one of the zeroes of ​the ​cubic ​polynomial $x^3+a{x}^2+bx+c$ is $-1$, ​then ​the product ​of the other two zeroes is

   1. (a)$b-a+1$
   2. (b)$b-a-1$
   3. (c)$a-b+1$
   4. (d)$a-b-1$
   Show answer

   (a) $b-a+1$

   Step-by-step solution →
10. Q10 Exemplar · Exercise 2.1
    The ​zeroes ​of the ​quadratic ​polynomial $x^2+99x+127$ are

    1. (a)both positive
    2. (b)both negative
    3. (c)one positive and one negative
    4. (d)both equal
    Show answer

    (b) both negative

    Step-by-step solution →
11. Q11 Exemplar · Exercise 2.1
    The ​zeroes ​of ​the ​quadratic polynomial $x^2+kx+k$, $k\ne 0$,

    1. (a)cannot both be positive
    2. (b)cannot both be negative
    3. (c)are always unequal
    4. (d)are always equal
    Show answer

    (a) cannot both be positive

    Step-by-step solution →
12. Q12 Exemplar · Exercise 2.1
    If the ​zeroes ​of the ​quadratic ​polynomial $a{x}^2+bx+c$, $c\ne 0$ ​are ​equal, ​then

    1. (a)$c$ and $a$ have opposite signs
    2. (b)$c$ and $b$ have opposite signs
    3. (c)$c$ and $a$ have the same sign
    4. (d)$c$ and $b$ have the same sign
    Show answer

    (c) $c$ and $a$ have the same sign

    Step-by-step solution →
13. Q13 Exemplar · Exercise 2.1
    If one ​of the zeroes of a quadratic polynomial of the form $x^2+ax+b$ is ​the ​negative ​of ​the other, then ​it

    1. (a)has no linear term and the constant term is negative.
    2. (b)has no linear term and the constant term is positive.
    3. (c)can have a linear term but the constant term is negative.
    4. (d)can have a linear term but the constant term is positive.
    Show answer

    (a) has no linear term and the constant term is negative.

    Step-by-step solution →
14. Q14 Exemplar · Exercise 2.1
    Which ​of ​the ​following is not ​the ​graph of a ​quadratic ​polynomial?

    1. (a)Graph (A)
    2. (b)Graph (B)
    3. (c)Graph (C)
    4. (d)Graph (D)
    Show answer

    (d) Graph (D)

    Step-by-step solution →
15. Q15 Exemplar · Exercise 2.2
    Answer the following and justify:

    1. (i)Can $x^2-1$ be the quotient on division of $x^6+2x^3+x-1$ by ​a polynomial in $x$ of degree 5?
    2. (ii)What will the quotient and remainder be on division of $a{x}^2+bx+c$ by $p{x}^3+q{x}^2+rx+s$, $p$ not equal to $0$?
    3. (iii)If on division of a polynomial $p(x)$ by a ​polynomial $g(x)$, the quotient is zero, what is the relation ​between the degrees of $p(x)$ ​and $g(x)$?
    4. (iv)If on division of a non-zero polynomial $p(x)$ by a polynomial $g(x)$, the remainder is zero, what is the relation between the degrees of $p(x)$ and $g(x)$?
    5. (v)Can the quadratic polynomial $x^2+kx+k$ have ​equal zeroes for some ​odd ​integer $k>1$?
    Show answer

    (i) No, degrees do not match ($5+2=7$ not 6). (ii) Quotient $=0$, remainder $=a{x}^2+bx+c$. (iii) $\mathrm{deg}p(x)<\mathrm{deg}g(x)$. (iv) $\mathrm{deg}p(x)\ge \mathrm{deg}g(x)$. (v) No - equal zeroes need $k=0$ or $4$, neither an odd integer greater than 1.

    Step-by-step solution →
16. Q16 Exemplar · Exercise 2.2
    Are the following statements 'True' or 'False'? Justify your answers.

    1. (i)If the zeroes ​of a quadratic polynomial $a{x}^2+bx+c$ are both positive, then $a$, $b$ and $c$ all have ​the same sign.
    2. (ii)If the graph ​of a polynomial ​intersects ​the $x$-axis at only one point, it cannot be a quadratic polynomial.
    3. (iii)If the graph of a polynomial intersects the $x$-axis at exactly two points, it need ​not be ​a quadratic polynomial.
    4. (iv)If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
    5. (v)If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.
    6. (vi)If all three zeroes of a cubic polynomial $x^3+a{x}^2-bx+c$ are positive, then at least one of $a$, $b$ and $c$ is non-negative.
    7. (vii)The only value of $k$ for which the ​quadratic polynomial $k{x}^2+x+k$ has equal zeros is $\frac{1}{2}$.
    Show answer

    (i) False, (ii) False, (iii) True, (iv) True, (v) True, (vi) False, (vii) False

    Step-by-step solution →
17. Q17 Exemplar · Exercise 2.3
    Find the zeroes of ​the ​following polynomials by factorisation method and verify the relations ​between the ​zeroes ​and the ​coefficients of the polynomials: 1. $4x^2-3x-1$ 2. $3x^2+4x-4$ 3. $5t^2+12t+7$ 4. $t^3-2t^2-15t$ 5. $2x^2+\frac{7}{2}x+\frac{3}{4}$ 6. $4x^2+5\sqrt{2}x-3$ ​7. $2s^2-(1+2\sqrt{2})s+\sqrt{2}$ 8. $v^2+4\sqrt{3}v-15$ 9. $y^2+\frac{3}{2}\sqrt{5}y-5$ ​10. $7y^2-\frac{11}{3}y-\frac{2}{3}$
    Show answer

    Zeroes: (1) -1/4, 1; (2) 2/3, -2; (3) -7/5, -1; (4) 0, 5, -3; (5) -3/2, -1/4; (6) $\sqrt{2}/4$, $-3\sqrt{2}/2$; (7) 1/2, $\sqrt{2}$; (8) $-5\sqrt{3}$, $\sqrt{3}$; (9) $-2\sqrt{5}$, $\sqrt{5}/2$; (10) -1/7, 2/3. All verified against the coefficient relations.

    Step-by-step solution →
18. Q18 Exemplar · Exercise 2.3
    Find ​the zeroes ​of the following polynomials ​by factorisation method ​and verify the ​relations between the zeroes ​and the coefficients of the ​polynomials: $3x^2+4x-4$
    Show answer

    The zeroes of $3x^2+4x-4$ are $\alpha =-2$ and $\beta =\frac{2}{3}$. Verified: sum of zeroes $=-2+\frac{2}{3}=-\frac{4}{3}=-\frac{b}{a}$, and product of zeroes $=(-2)\times \frac{2}{3}=-\frac{4}{3}=\frac{c}{a}$.

    Step-by-step solution →
19. Q19 Exemplar · Exercise 2.3
    Find ​the zeroes of the ​following ​polynomials by ​factorisation method and verify ​the relations between the zeroes and the ​coefficients of the polynomials: $5t^2+12t+7$
    Show answer

    The zeroes of $5t^2+12t+7$ are $\alpha =-1$ and $\beta =-\frac{7}{5}$. Verified: sum of zeroes $=-1-\frac{7}{5}=-\frac{12}{5}=-\frac{b}{a}$, and product of zeroes $=(-1)\times \left(-\frac{7}{5}\right)=\frac{7}{5}=\frac{c}{a}$.

    Step-by-step solution →
20. Q20 Exemplar · Exercise 2.3
    Find ​the zeroes of ​the ​following polynomials by factorisation method ​and verify the relations ​between ​the zeroes and ​the coefficients of the ​polynomials: $t^3-2t^2-15t$
    Show answer

    The zeroes of $t^3-2t^2-15t$ are $t=0$, $t=5$ and $t=-3$. Verified: $\alpha +\beta +\gamma =2=-\frac{b}{a}$, $\alpha \beta +\beta \gamma +\gamma \alpha =-15=\frac{c}{a}$, and $\alpha \beta \gamma =0=-\frac{d}{a}$.

    Step-by-step solution →