NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

Komal MiglaniUpdated on 22 Aug 2025, 08:42 AM IST

Have you ever noticed how the path of a roller coaster, the trajectory of a football, or the economic trend predictions follow a certain pattern, that is the power of polynomials. Polynomials are not just some algebraic expression; they are one of the main pillars of mathematics. According to the latest syllabus, this chapter covers the basic concepts of polynomials, including the degree of Polynomials, Zeroes of a Polynomial, the Geometrical Meaning of the Zeroes of a Polynomial, and the Relationship between Zeroes and Coefficients of a Polynomial. Understanding these concepts will make students more efficient in solving problems involving polynomials and will also build a strong foundation for advanced polynomial concepts. NCERT Solutions for Class 10 can help the students immensely‌.

This Story also Contains

  1. NCERT Solutions for Class 10 Maths Chapter 2 Polynomials PDF Free Download
  2. NCERT Solutions for Class 10 Maths Chapter 2: Exercise Questions
  3. Polynomials Class 10 Solutions - Exercise Wise
  4. Polynomials Class 10 Chapter 2: Topics
  5. NCERT Solutions for Class 10 Maths Chapter 2 Polynomials: Important Formulae
  6. NCERT Solutions for Class 10 Maths: Chapter Wise
NCERT Solutions for Class 10 Maths Chapter 2 Polynomials
polynomials

This NCERT Solutions for class 10 Maths article about Polynomials is designed by our experienced subject experts at Careers360 to offer a systematic and structured approach to these important concepts. These solutions also help students prepare well for exams and gain knowledge about the various natural processes occurring around them through a series of solved questions provided in the NCERT textbook exercises. It covers questions from all the topics and will help you improve your speed and accuracy. Many toppers rely on NCERT Solutions since they are designed as per the latest syllabus. Get all solved exercises, full syllabus notes, and a free PDF from the NCERT article.

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials PDF Free Download

Students who wish to access the NCERT solutions for Class 10 Chapter 2 can click on the link below to download the entire solution in PDF.

Download Solution PDF

NCERT Solutions for Class 10 Maths Chapter 2: Exercise Questions

Polynomials Class 10 Exercise: 2.1
Total Questions: 1
Page number: 18
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Q1 (1): The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the numbers of zeroes of p(x), in each case.

1635918537229

Answer: The number of zeroes of p(x) is zero as the curve does not intersect the x-axis.

Q1 (2): 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

1635918562539

Answer: The number of zeroes of p(x) is one as the curve intersects the x-axis only once.

Q1 (3): 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

1635918572380

Answer: The number of zeroes of p(x) is three as the graph intersects the x-axis thrice.

Q1 (4): 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

1635918582058

Answer: The number of zeroes of p(x) is two as the graph intersects the x-axis twice.

Q1 (5): 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

1635918596067

Answer: The number of zeroes of p(x) is four as the graph intersects the x-axis four times.

Q1 (6): 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

1635918606641

Answer: The number of zeroes of p(x) is three as the graph intersects the x-axis thrice.

Polynomials Class 10 Exercise: 2.2
Total Questions: 2
Page number: 23

Q1 (i): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. $x^2-2 x-8$

Answer:

x 2 - 2x - 8 = 0

x 2 - 4x + 2x - 8 = 0

x(x-4) +2(x-4) = 0

(x+2)(x-4) = 0

The zeroes of the given quadratic polynomial are -2 and 4

$\\\alpha =-2\\, \beta =4$

VERIFICATION:

Sum of roots:

$
\begin{aligned}
& \alpha+\beta=-2+4=2 \\
& -\frac{\text { coefficient of } x}{\text { coefficient of } x^2} \\
& =-\frac{-2}{1} \\
& =2 \\
& =\alpha+\beta
\end{aligned}
$


Verified
Product of roots:

$
\begin{aligned}
& \alpha \beta=-2 \times 4=-8 \\
& \frac{\text { constant term }}{\text { coefficient of } x^2} \\
& =\frac{-8}{1} \\
& =-8 \\
& =\alpha \beta
\end{aligned}
$

Verified

Q1 (ii): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. $4 s^2-4 s+1$

Answer:

$
\begin{aligned}
& 4 s^2-4 s+1=0 \\
& 4 s^2-2 s-2 s+1=0 \\
& 2 s(2 s-1)-1(2 s-1)=0 \\
& (2 s-1)(2 s-1)=0
\end{aligned}
$
The zeroes of the given quadratic polynomial are $1 / 2$ and $1 / 2$

$
\begin{aligned}
& \alpha=\frac{1}{2} \\
& \beta=\frac{1}{2}
\end{aligned}
$
VERIFICATION
Sum of roots:

$
\alpha+\beta=\frac{1}{2}+\frac{1}{2}=1
$

$
\begin{aligned}
& -\frac{\text { coefficient of } x}{\text { coefficient of } x^2} \\
& =-\frac{-4}{4} \\
& =1 \\
& =\alpha+\beta
\end{aligned}
$
Verified
Product of roots:

$
\begin{aligned}
& \alpha \beta=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4} \\
& \frac{\text { constant term }}{\text { coefficient of } x^2} \\
& =\frac{1}{4} \\
& =\alpha \beta
\end{aligned}
$

Verified

Q1 (iii): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. $6 x^2-3-7 x$

Answer:

6x 2 - 3 - 7x = 0

6x 2 - 7x - 3 = 0

6x 2 - 9x + 2x - 3 = 0

3x(2x - 3) + 1(2x - 3) = 0

(3x + 1)(2x - 3) = 0

The zeroes of the given quadratic polynomial are -1/3 and 3/2

$
\begin{aligned}
& \alpha=-\frac{1}{3} \\
& \beta=\frac{3}{2}
\end{aligned}
$
Sum of roots:

$
\begin{aligned}
& \alpha+\beta=-\frac{1}{3}+\frac{3}{2}=\frac{7}{6} \\
& -\frac{\text { coefficient of } x}{\text { coefficient of } x^2} \\
& =-\frac{-7}{6} \\
& =\frac{7}{6} \\
& =\alpha+\beta
\end{aligned}
$

Verified

Product of roots:

$\begin{aligned} & \alpha \beta=-\frac{1}{3} \times \frac{3}{2}=-\frac{1}{2} \\ & \frac{\text { constant term }}{\text { coefficient of } x^2} \\ & =\frac{-3}{6} \\ & =-\frac{1}{2} \\ & =\alpha \beta\end{aligned}$

Verified

Q1 (iv): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. $4 u^2+8 u$

Answer:
4u 2 + 8u = 0

4u(u + 2) = 0

The zeroes of the given quadratic polynomial are 0 and -2

$
\begin{aligned}
& \alpha=0 \\
& \beta=-2
\end{aligned}
$
VERIFICATION:
Sum of roots:

$
\begin{aligned}
& \alpha+\beta=0+(-2)=-2 \\
& -\frac{\text { coefficient of } x}{\text { coefficient of } x^2} \\
& =-\frac{8}{4} \\
& =-2 \\
& =\alpha+\beta
\end{aligned}
$


Verified
Product of roots:

$
\alpha \beta=0 \times-2=0
$

$\begin{aligned} & \frac{\text { constant term }}{\text { coeff ficient of } x^2} \\ & =\frac{0}{4} \\ & =0 \\ & =\alpha \beta\end{aligned}$

Verified

Q1 (v): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. $t^2-15$

Answer:
t 2 - 15 = 0

$
(t-\sqrt{15})(t+\sqrt{15})=0
$


The zeroes of the given quadratic polynomial are $-\sqrt{15}$ and $\sqrt{15}$

$
\begin{aligned}
& \alpha=-\sqrt{15} \\
& \beta=\sqrt{15}
\end{aligned}
$
VERIFICATION:
Sum of roots:

$
\begin{aligned}
& \alpha+\beta=-\sqrt{15}+\sqrt{15}=0 \\
& -\frac{\text { coefficient of } x}{\text { coefficient of } x^2} \\
& =-\frac{0}{1} \\
& =0 \\
& =\alpha+\beta
\end{aligned}
$

Verified

Product of roots:

$\begin{aligned} & \alpha \beta=-\sqrt{15} \times \sqrt{15}=-15 \\ & \frac{\text { constant term }}{\text { coefficient of } x^2} \\ & =\frac{-15}{1} \\ & =-15 \\ & =\alpha \beta\end{aligned}$

Verified

Q1 (vi): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. $3 x^2-x-4$

Answer:
3x 2 - x - 4 = 0

3x 2 + 3x - 4x - 4 = 0

3x(x + 1) - 4(x + 1) = 0

(3x - 4)(x + 1) = 0

The zeroes of the given quadratic polynomial are 4/3 and -1

$
\begin{aligned}
& \alpha=\frac{4}{3} \\
& \beta=-1
\end{aligned}
$
VERIFICATION:
Sum of roots:

$
\begin{aligned}
& \alpha+\beta=\frac{4}{3}+(-1)=\frac{1}{3} \\
& -\frac{\text { coefficient of } x}{\text { coefficient of } x^2} \\
& =-\frac{-1}{3} \\
& =\frac{1}{3} \\
& =\alpha+\beta
\end{aligned}
$

Verified

Product of roots:

$\begin{aligned} & \alpha \beta=\frac{4}{3} \times-1=-\frac{4}{3} \\ & \frac{\text { constant term }}{\text { coefficient of } x^2} \\ & =\frac{-4}{3} \\ & =\alpha \beta\end{aligned}$

Verified

Q2 (i): Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively. 1/4 , -1

Answer:

$
\begin{aligned}
& \alpha+\beta=\frac{1}{4} \\
& \alpha \beta=-1
\end{aligned}
$
The required quadratic polynomial is

$
\begin{aligned}
& x^2-(\alpha+\beta)x+\alpha \beta=0 \\
& x^2-\frac{1}{4} x-1=0 \\
& 4 x^2-x-4=0
\end{aligned}
$

Q2 (ii): Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively. $\sqrt{2}, 1 / 3$

Answer:

$
\begin{aligned}
& \alpha+\beta=\sqrt{2} \\
& \alpha \beta=\frac{1}{3} \\
& x^2-(\alpha+\beta)x+\alpha \beta=0 \\
& x^2-\sqrt{2} x+\frac{1}{3}=0 \\
& 3 x^2-3 \sqrt{2} x+1=0
\end{aligned}
$
The required quadratic polynomial is $3 x^2-3 \sqrt{2} x+1$

Q2 (iii): Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively. $0, \sqrt{5}$

Answer:

$\begin{aligned} & \alpha+\beta=0 \\ & \alpha \beta=\sqrt{5} \\ & x^2-(\alpha+\beta)x+\alpha \beta=0 \\ & x^2-0 x+\sqrt{5}=0 \\ & x^2+\sqrt{5}=0\end{aligned}$

The required quadratic polynomial is x 2 + $\sqrt{5}$ .

Q2 (iv): Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively. 1,1

Answer:

$\begin{aligned} & \alpha+\beta=1 \\ & \alpha \beta=1 \\ & x^2-(\alpha+\beta)x+\alpha \beta=0 \\ & x^2-1 x+1=0 \\ & x^2-x+1=0\end{aligned}$

The required quadratic polynomial is x 2 - x + 1

Q2 (v): Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively. $-\frac{1}{4}, \frac{1}{4}$

Answer:

$\begin{aligned} & \alpha+\beta=-\frac{1}{4} \\ & \alpha \beta=\frac{1}{4} \\ & x^2-(\alpha+\beta)x+\alpha \beta=0 \\ & x^2-\left(-\frac{1}{4}\right) x+\frac{1}{4}=0 \\ & 4 x^2+x+1=0\end{aligned}$

The required quadratic polynomial is 4x 2 + x + 1

Q2 (vi): Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively. 4,1

Answer:

$\begin{aligned} & \alpha+\beta=4 \\ & \alpha \beta=1 \\ & x^2-(\alpha+\beta)x+\alpha \beta=0 \\ & x^2-4 x+1=0\end{aligned}$

The required quadratic polynomial is x 2 - 4x + 1.

Polynomials Class 10 Chapter 2: Topics

Topics you will learn in NCERT Class 10 Maths Chapter 2 Polynomials include:

  • 2.1 Introduction
  • 2.2 Geometrical Meaning of the Zeros of a Polynomial
  • 2.3 Relationship between Zeros and Coefficients of a Polynomial

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials: Important Formulae

Polynomials

A polynomial $p(x)$ is an algebraic expression that can be written in the form of

$
p(x)=a_n x^n+\ldots+a_2 x^2+a_1 x+a_0
$

Here $a_0, a_1, a_2, \ldots, a_n$ are real numbers and each power of x is a non-negative integer.

Each real number ai is called a coefficient. The number a0  that is not multiplied by a variable is called a constant. Each product  $a_i x_i$  is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called the degree of the polynomial. The leading term is the term with the highest power, and its coefficient is called the leading coefficient.

Types of Polynomials

The types of polynomials based on the number of terms are

  • Monomial: A monomial is a polynomial with one term. Eg. $3x$
  • Binomial: A binomial is a polynomial with two terms. Eg. $3x+2y$
  • Trinomial: A trinomial is a polynomial with three terms. Eg. $4x^2+ 3x+2y$
  • Multinomial: A general term for polynomials with more than three terms. Eg. $7x^5+ 5x^3+3x^2+2x=1$
  • Constant Polynomial: A constant polynomial is a polynomial with no variable terms but with only a constant term. Eg. $P(x) = 5$
  • Zero Polynomial: A polynomial with coefficients as zero. Eg. $0x^2+0x, 0$

Types of polynomials (based on the degree of a polynomial)

  • Linear Polynomial: A polynomial with degree one. Eg. $3x+5y = 5$
  • Quadratic Polynomial: A Polynomial with degree two. any quadratic polynomial in $x$ is of the form $ax^2 + bx + c$, where $a, b, c$ are real numbers and $a \neq 0$. E.g. $2x^2+3x+2=0$
  • Cubic Polynomial: A Polynomial with degree three. The general form of a cubic polynomial is $ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are real numbers and $a \neq 0 $. E.g. $5x^3+3x^2+2x=1$
  • Higher-degree polynomial: Polynomials with a degree of more than three. E.g. $7x^5+5x^3+3$

Zeros of a Polynomial

If a real number $k$ satisfies the given polynomial, then $k$ is a zero of that polynomial. (i.e) A real number k is the zero of the polynomial $P(x)$, if $P(k) = 0$

Example: Let $P(x) = x^2 -4$. Let $x = 2$, then $P(x) = 2^2 -4 = 4-4=0$. Therefore, $2$ is the zero of the polynomial $P(x)$.

Graphical Representation of Zeros of a Polynomial

For a polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at most n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.

The number of zeros of a polynomial can be found by the number of points of the graph of the polynomial intersecting the x-axis.

Relationship Between Zeros and Coefficients of the Polynomial

Linear Polynomial:

The zero of the linear polynomial $ax+b$ = $-\frac{b}{a}$.

Quadratic Polynomial:

For the quadratic polynomial $ax^2+bx+c=0$ with zeros $x_1$ and $x_2$,

Sum of zeros, $x_1+x_2= -\frac{b}{a}$

Product of zeros $x_1 x_2= \frac{c}{a}$

Cubic Polynomial:

For the quadratic polynomial $ax^3+bx^2+cx+d=0$ with zeros $x_1$, $x_2$ and $x_3$,

Sum of zeros, $x_1+x_2= -\frac{b}{a}$

Sum of product of two zeros, $x_1 x_2+x_2 x_3+x_3 x_1= \frac{c}{a}$

Product of zeros $x_1 x_2= -\frac{d}{a}$

NCERT Solutions for Class 10 Maths: Chapter Wise

We at Careers360 compiled all the NCERT class 10 Maths solutions in one place for easy student reference. The following links will allow you to access them.

NCERT Solutions of Class 10 - Subject Wise

Students can use the following link to check the solutions of science-related questions in the NCERT book in depth.

NCERT Exemplar solutions - Subject-wise

After completing the NCERT textbooks, students should practice exemplar exercises for a better understanding of the chapters and clarity. The following links will help students find exemplar exercises.

NCERT Books and NCERT Syllabus here

Here are some useful links for NCERT books and the NCERT syllabus for class 10:

Frequently Asked Questions (FAQs)

Q: What are the polynomials in class 10 NCERT?
A:

A polynomial p(x) is an algebraic expression that can be written in the form of

p(x)=anxn++a2x2+a1x+a0

Here a0,a1,a2,,an are real numbers and each power of x is a non-negative integer.

Q: How to find the relationship between zeros and coefficients of a quadratic polynomial?
A:

Relationship between zeros and coefficients of a quadratic polynomial

For the quadratic polynomial ax2+bx+c=0 with zeros x1 and x2,

Sum of zeros, x1+x2=b/a

Product of zeros x1x2=c/a

Q: What is the difference between linear, quadratic and cubic polynomial?
A:

The difference between linear, quadratic and cubic polynomials is the degree of the polynomial. The degree of the linear polynomial is one, the degree of the quadratic polynomial is two, and the degree of the cubic polynomial is three.

Q: How many types of polynomials are there?
A:

Based on the number of terms, polynomials are of 4 types, monomial, binomial, trinomial and multinomial. 

Based on the degree, polynomials are of 4 types, namely, linear, quadratic, cubic and higher-degree polynomials.

Q: How to find the number of zeroes of a polynomial graphically?
A:

For a polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at most n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.

The number of zeros of a polynomial can be found by the number of points of the graph of the polynomial intersecting with the x-axis. 

Q: What is the degree of a polynomial?
A:

The highest power of the variable that occurs in the polynomial is called the degree of a polynomial.

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Yes, you can give the CBSE board exam in 2027.

If your date of birth is 25.05.2013, then in 2027 you will be around 14 years old, which is the right age for Class 10 as per CBSE rules. So, there is no problem.

Hope it helps !

Hello! If you selected “None” while creating your APAAR ID and forgot to mention CBSE as your institution, it may cause issues later when linking your academic records or applying for exams and scholarships that require school details. It’s important that your APAAR ID correctly reflects your institution to avoid verification problems. You should log in to the portal and update your profile to select CBSE as your school. If the system doesn’t allow editing, contact your school’s administration or the APAAR support team immediately so they can correct it for you.

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Your school is the best place to get this information.

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It appears you are asking if you can fill out a form after passing your 10th grade examination in the 2024-2025 academic session.

The answer depends on what form you are referring to. Some forms might be for courses or examinations where passing 10th grade is a prerequisite or an eligibility criteria, such as applying for further education or specific entrance exams. Other forms might be related to other purposes, like applying for a job, which may also have age and educational requirements.

For example, if you are looking to apply for JEE Main 2025 (a competitive exam in India), having passed class 12 or appearing for it in 2025 are mentioned as eligibility criteria.

Let me know if you need imformation about any exam eligibility criteria.

good wishes for your future!!

Hello Aspirant,

"Real papers" for CBSE board exams are the previous year's question papers . You can find these, along with sample papers and their marking schemes , on the official CBSE Academic website (cbseacademic.nic.in).

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