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 3 of 3

1. Q41 CBSE 2026
   The ​sum and ​product of ​zeroes of ​a quadratic ​polynomial $p(x)$ are $-\frac{1}{3}$ and $2$ ​respectively. The polynomial $p(x)$ is ​:

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

   (d) $-3x^2-x-6$

   Step-by-step solution →
2. Q42 CBSE 2026
   In ​the following question, a statement ​of Assertion (A) is followed by a statement ​of ​Reason ​(R). ​Choose ​the correct ​option :

   Assertion (A): The polynomial $p(y)=y^2+4y+3$ has two zeroes.

   Reason (R): A quadratic polynomial can have at most two zeroes.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
   3. (c)Assertion (A) is true, but Reason (R) is false.
   4. (d)Assertion (A) is false, but Reason (R) is true.
   Show answer

   (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).

   Step-by-step solution →
3. Q43 CBSE 2026
   If $\alpha$, $\beta$ ​are ​the ​zeroes ​of ​the polynomial $p(x)=x^2-3x-1$, ​then find the value of $\frac{1}{\alpha }+\frac{1}{\beta }$.
   Show answer

   The value of $\frac{1}{\alpha }+\frac{1}{\beta }$ is $-3$.

   Step-by-step solution →
4. Q44 CBSE 2026
   If $\alpha ,\beta$ are the zeroes ​of the quadratic ​polynomial $p{x}^2+qx+r$, ​then ​find ​the ​value ​of ${\alpha }^3\beta +{\beta }^3\alpha$.
   Show answer

   The value of ${\alpha }^3\beta +{\beta }^3\alpha$ is $\frac{r(q^2-2pr)}{p^3}$.

   Step-by-step solution →
5. Q45 CBSE 2026
   Find the ​quadratic ​polynomial ​the sum of whose ​zeroes is $1$ and their ​product is $-12$. Hence find the ​zeroes of the polynomial.
   Show answer

   The quadratic polynomial is $x^2-x-12$, and its zeroes are $4$ and $-3$.

   Step-by-step solution →
6. Q46 CBSE 2026
   Verify ​the relation ​between ​the ​zeroes and the coefficients of the ​quadratic ​polynomial $4x^2-9$.
   Show answer

   The zeroes of $4x^2-9$ are $\frac{3}{2}$ and $-\frac{3}{2}$. Sum of zeroes $=0=-\frac{b}{a}$ and product of zeroes $=-\frac{9}{4}=\frac{c}{a}$. Hence the relationship between the zeroes and the coefficients is verified.

   Step-by-step solution →
7. Q47 CBSE 2026
   Verify the relationship between ​the zeroes and ​the coefficients ​of the ​polynomial $p(x)=3x^2-5x$.
   Show answer

   The zeroes of $p(x)=3x^2-5x$ are $0$ and $\frac{5}{3}$ ; sum of zeroes $=\frac{5}{3}=-\frac{b}{a}$ and product of zeroes $=0=\frac{c}{a}$. Hence the relationship between the zeroes and the coefficients is verified.

   Step-by-step solution →
8. Q48 CBSE 2026
   Find the ​zeroes ​of the ​polynomial $p(x)=15x^2-19x+6$.
   Show answer

   The zeroes are $x=\frac{2}{3}$ and $x=\frac{3}{5}$.

   Step-by-step solution →
9. Q49 CBSE 2026
   If $\alpha ,\beta$ ​are the ​zeroes ​of ​the polynomial $p(x)=6x^2-5x-3$, ​then find ​the value ​of $\frac{1}{\alpha }+\frac{1}{\beta }$.
   Show answer

   $\frac{1}{\alpha }+\frac{1}{\beta }=-\frac{5}{3}$

   Step-by-step solution →
10. Q50 CBSE 2026
    One zero ​of a ​quadratic ​polynomial is twice ​the ​other. If the sum ​of zeroes is $(-6)$, ​find the polynomial.
    Show answer

    The required polynomial is $x^2+6x+8$ (or any non-zero multiple of it)

    Step-by-step solution →
11. Q51 CBSE 2026
    If one zero ​of the ​polynomial $p(x)=(k-1)x^2-(4k+1)x+10$ is $5$, ​find the ​value ​of $k$.
    Show answer

    $k=4$.

    Step-by-step solution →
12. Q52 CBSE 2026
    One ​zero of the polynomial $4x^2-12x+(2k+1)$ ​is five times ​the ​other. ​Find ​the value of $k$.
    Show answer

    $k=2$.

    Step-by-step solution →
13. Q53 CBSE 2026
    $\alpha$ ​and $\beta$ are ​the zeroes of ​the polynomial $5x^2-16x-10$. ​Find ​the ​value of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$.
    Show answer

    $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=-\frac{178}{25}$ (which equals $-7.12$).

    Step-by-step solution →
14. Q54 CBSE 2026
    If $\alpha ,\beta$ are zeroes ​of ​the polynomial $p(x)=5x^2-7x-3$, then form ​a quadratic polynomial whose ​zeroes ​are $\frac{2}{\alpha }$ ​and $\frac{2}{\beta }$.
    Show answer

    A required polynomial is $3x^2+14x-20$ (any non-zero multiple is also acceptable).

    Step-by-step solution →
15. Q55 CBSE 2026
    Find the zeroes ​of ​the polynomial $p(x)=3x^2+7x-20$ ​and ​verify the relationship between ​its zeroes ​and ​the ​coefficients.
    Show answer

    The zeroes are $x=\frac{5}{3}$ and $x=-4$. Verification: sum $=\frac{5}{3}-4=-\frac{7}{3}=-\frac{b}{a}$ and product $=\frac{5}{3}\times (-4)=-\frac{20}{3}=\frac{c}{a}$.

    Step-by-step solution →
16. Q56 CBSE 2026
    During a theatre drama, a backdrop of ​building ​arches was used. The shape ​of the curve shown below ​can be represented by the polynomial $p(x)=-x^2+2x+8$, where $x$ is the length (in feet) ​on ​stage level. Based on the ​figure given above, answer the following questions:

    1. (i)Determine the height of the arch.[1]
    2. (ii)(a)Find zeroes of the polynomial $p(x)$. Which points on the graph represent the zeroes?[2]
    3. (ii)(b)(OR) Find the span of the arch on the stage floor.[2]
    4. (iii)Write the coordinates of the point of intersection of the above curve with the y-axis.[1]
    Show answer

    (i) The height of the arch is $9$ feet (top of the arch $C(1,9)$). (ii)(a) The zeroes are $x=4$ and $x=-2$, represented by the points $A(4,0)$ and $B(-2,0)$ where the curve meets the x-axis. (ii)(b) The span of the arch on the floor is $6$ feet. (iii) The curve meets the y-axis at $(0,8)$.

    Step-by-step solution →
17. Q57 CBSE 2026
    An arch of ​a railway bridge, built on the Chenab riverbed, is ​shown in the diagram. ​It ​is a parabolic arch connecting two hills ​at $P$ and $Q$. ​The parabolic curve is represented by the polynomial $p(x)=-0.0025x^2-0.025x+136$. Observe the diagram and, based on the above information, ​answer the following questions :

    1. (i)Write the co-ordinates of point $A$.[1]
    2. (ii)Find the span of the arch.[1]
    3. (iii) (a)Write the zeroes of the polynomial using the diagram and verify the relationship between the sum of the zeroes and the coefficients of the polynomial.[2]
    4. (iii) (b)Find the values of $p(x)$ at $x=100$ and $x=-100$. Are they the same?[2]
    Show answer

    (i) $A=(0,136)$. (ii) The span of the arch is $467$ m. (iii) (a) The zeroes are $x=-238.5$ and $x=228.5$; their sum $-10$ equals $-\frac{b}{a}=-\frac{-0.025}{-0.0025}=-10$, so the relationship holds. (iii) (b) $p(100)=108.5$ and $p(-100)=113.5$; no, they are not the same.

    Step-by-step solution →
18. Q58 CBSE 2026
    While playing badminton, ​Ravi has set the barrier chain ​hung ​between two posts at the edge of the walkway of a street. ​It ​is hung in the shape of a parabola. Based ​on the ​above information, ​answer the following questions:

    1. (a)Which type of the polynomial (linear, quadratic, cubic etc.) is graphically represented by a parabola?[1]
    2. (b)If the polynomial represented by a parabola intersects the $x$-axis at $-2$ and $3$ and the $y$-axis at $-3$, then write the zeroes of the parabola.[1]
    3. (c)Find the expression for the above polynomial. OR If the zeroes of the polynomial are $-5$ and $3$, find its expression.[2]
    Show answer

    (a) A quadratic polynomial (degree $2$). (b) The zeroes are $-2$ and $3$. (c) $p(x)=\frac{1}{2}(x+2)(x-3)=\frac{1}{2}{x}^2-\frac{1}{2}x-3$. OR, for zeroes $-5$ and $3$, $p(x)=x^2+2x-15$.

    Step-by-step solution →