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NCERT Solutions for Class 10 Maths Chapter 2 Polynomials
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CBSE Class 10th Exam Date:15 Jul' 2025 - 15 Jul' 2025
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. As per the latest syllabus, this chapter contains the basic concepts of polynomials like Degree of Polynomials, Zeroes of a Polynomial, Geometrical Meaning of the Zeroes of a Polynomial, and 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 in this regard.
This NCERT Solutions for class 10 Maths article about Polynomials is designed by our experienced subject experts to offer a systematic and structured approach to these important concepts and help students to prepare well for exams and to gain knowledge about all the natural processes happening around them by a series of solved questions given in the NCERT textbook exercise. It covers questions from all the topics and will help you improve your speed and accuracy. The solutions for other subjects and classes can be downloaded from the NCERT Solutions.
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.
NCERT Solutions for Class 10 Maths Chapter 2: Exercise Questions
| Polynomials Class 10 Exercise: 2.1 Total Questions: 1 Page number: 18 |
Answer: The number of zeroes of p(x) is zero as the curve does not intersect the x-axis.
Answer: The number of zeroes of p(x) is one as the curve intersects the x-axis only once.
Answer: The number of zeroes of p(x) is three as the graph intersects the x-axis thrice.
Answer: The number of zeroes of p(x) is two as the graph intersects the x-axis twice.
Answer: The number of zeroes of p(x) is four as the graph intersects the x-axis four times.
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 |
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
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
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
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
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
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
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}
$
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 $3 x^2-3 \sqrt{2} x+1$
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 + $\sqrt{5}$ .
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
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
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.
Polynomial Class 10 Solutions - Exercise Wise
Here are the exercise-wise links for the NCERT class 10 chapter 2 Polynomial:
Polynomials Class 10 Chapter 2: Topics
The important topics covered in the NCERT Class 10 Maths, chapter 2, Polynomials are:
- 2.1 Introduction
- 2.2 Geometrical Meaning of the Zeros of a
- Polynomial
- 2.3 Relationship between Zeros and Coefficients
- of a Polynomial
Recommended eBooks
Related eBooks & Sample Papers
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: 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 $a \neq 0$. Eg. $2x^2+3x+2=0$
- Cubic Polynomial: 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 $. Eg. $5x^3+3x^2+2x=1$
- Higher-degree polynomial: Polynomials with degree more than three. Eg. $7x^5+5x^3+3$
Zeros of a Polynomial
If a real number $k$ satisfies the given polynomial, then $k$ is the 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 $x_1 x_2= -\frac{d}{a}$
CBSE Class 10th Syllabus: Subjects & Chapters
Select your preferred subject to view the chapters
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 links to check the solution of maths and science-related questions 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 to 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
$p(x)=a_nx^n+…+a_2x^2+a_1x+a_0$
Here $a_0,a_1,a_2,…,a_n$ 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 $ax^2+bx+c=0$ with zeros $x_1$ and $x_2$,
Sum of zeros, $x_1+x_2=−b/a$
Product of zeros $x_1x_2=c/a$
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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Hello
Since you are a domicile of Karnataka and have studied under the Karnataka State Board for 11th and 12th , you are eligible for Karnataka State Quota for admission to various colleges in the state.
1. KCET (Karnataka Common Entrance Test): You must appear for the KCET exam, which is required for admission to undergraduate professional courses like engineering, medical, and other streams. Your exam score and rank will determine your eligibility for counseling.
2. Minority Income under 5 Lakh : If you are from a minority community and your family's income is below 5 lakh, you may be eligible for fee concessions or other benefits depending on the specific institution. Some colleges offer reservations or other advantages for students in this category.
3. Counseling and Seat Allocation:
After the KCET exam, you will need to participate in online counseling.
You need to select your preferred colleges and courses.
Seat allocation will be based on your rank , the availability of seats in your chosen colleges and your preferences.
4. Required Documents :
Domicile Certificate (proof that you are a resident of Karnataka).
Income Certificate (for minority category benefits).
Marksheets (11th and 12th from the Karnataka State Board).
KCET Admit Card and Scorecard.
This process will allow you to secure a seat based on your KCET performance and your category .
check link for more details
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Hope this helps you .
Hello Aspirant, Hope your doing great, your question was incomplete and regarding what exam your asking.
Yes, scoring above 80% in ICSE Class 10 exams typically meets the requirements to get into the Commerce stream in Class 11th under the CBSE board . Admission criteria can vary between schools, so it is advisable to check the specific requirements of the intended CBSE school. Generally, a good academic record with a score above 80% in ICSE 10th result is considered strong for such transitions.
hello Zaid,
Yes, you can apply for 12th grade as a private candidate .You will need to follow the registration process and fulfill the eligibility criteria set by CBSE for private candidates.If you haven't given the 11th grade exam ,you would be able to appear for the 12th exam directly without having passed 11th grade. you will need to give certain tests in the school you are getting addmission to prove your eligibilty.
best of luck!
According to cbse norms candidates who have completed class 10th, class 11th, have a gap year or have failed class 12th can appear for admission in 12th class.for admission in cbse board you need to clear your 11th class first and you must have studied from CBSE board or any other recognized and equivalent board/school.
You are not eligible for cbse board but you can still do 12th from nios which allow candidates to take admission in 12th class as a private student without completing 11th.
Popular CBSE Class 10th Questions
A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is
| Option 1)
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Option 2)
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| Option 3)
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Option 4)
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An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range
| Option 1)
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Option 2)
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| Option 3)
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Option 4)
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A particle is projected at 600 to the horizontal with a kinetic energy . The kinetic energy at the highest point
| Option 1)
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Option 2)
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| Option 3)
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Option 4)
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In the reaction,
| Option 1)
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Option 2)
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| Option 3)
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Option 4)
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How many moles of magnesium phosphate, will contain 0.25 mole of oxygen atoms?
| Option 1)
0.02 |
Option 2)
3.125 × 10-2 |
| Option 3)
1.25 × 10-2 |
Option 4)
2.5 × 10-2 |
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