Have you ever noticed how the path of a roller coaster, the trajectory of a football, or 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 enable students to solve problems involving polynomials more efficiently and build a strong foundation for advanced polynomial concepts. NCERT Solutions for Class 10 can help the students immensely.
This Story also Contains
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 master polynomials in detail. 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.
Careers360 brings you NCERT Class 10 Maths Chapter 2 Polynomials solutions, carefully prepared by subject experts to simplify your studies and help in exams. A downloadable PDF is available — click the link below to access it.
Below are the detailed NCERT Class 10 Maths Chapter 2 Polynomials question answers provided in the textbook.
Polynomials Class 10 Question Answers: 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 Question Answers: Exercise: 2.2 Total Questions: 2 Page number: 23 |
Answer:
x2 - 2x - 8 = 0
x2 - 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:
6x2 - 3 - 7x = 0
6x2 - 7x - 3 = 0
6x2 - 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:
4u2 + 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:
t2 - 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:
3x2 - x - 4 = 0
3x2 + 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 x2 - 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 4x2 + 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 x2 - 4x + 1.
Exercise-wise NCERT Solutions of Polynomials Class 10 Maths Chapter 2 are provided in the link below.
Topics you will learn in NCERT Class 10 Maths Chapter 2 Polynomials include:
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.
The types of polynomials based on the number of terms are:
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 $P(x)$.
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.
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 $x_1 x_2= -\frac{d}{a}$
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.
Also, read,
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.
Here are some useful links for NCERT books and the NCERT syllabus for class 10:
Frequently Asked Questions (FAQs)
Polynomials are used in physics, engineering, economics, and computer science to model curves, solve equations, and optimise solutions.
NCERT Class 10 Maths Chapter 2 contains 2 exercises, including examples.
The highest power of the variable that occurs in the polynomial is called the degree of the 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.
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.
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.
On Question asked by student community
Hello
Visit the official website of the Rajasthan Education Department or the Shala Darpan portal.
Look for the “Latest News” or “Examination” section. Check school notice boards or ask your class teacher for updates.
Some district education office websites may also upload the key. Avoid unofficial websites as they may provide incorrect or fake keys.
Hrllo,
If you want to view the CM Shri School 2025-26 admission test result, visit the official website http://www.edudel.nic.in/ (http://www.edudel.nic.in/,) ; here, click on the "CM Shri Schools Admission Test - 2025." then select the "CM Shri Admission Test Result 2025 - Merit List" link. Here you need to log in with your credentials and view or download the merit list pdf.
I hope it will clear your query!!
Hello aspirant,
Fairness and equal opportunity are guaranteed by the rigorous merit-based admissions process at CM Shri Schools. The merit list will be made public on edudel.nic.in, the official DOE website. Based on how well they did on the admission exam, students will be shortlisted.
To know the result, you can visit our site through following link:
https://school.careers360.com/articles/cm-shri-school-admission-test-result-2025
Thank you
Hello
The CM Shri School Admission Test was conducted for students seeking quality education.
The result and selection list are expected to be released around 20 September 2025.
Students who qualified will get admission into CM Shri Model Schools across the state.
The official list of selected students will be available on the edudel.nic.in website.
Candidates can check their names in the merit list PDF once uploaded.
Selection is based on the student’s performance in the entrance exam.
Keep checking the official site or the school notice board for timely updates.
You can check CM Shri school result here: https://school.careers360.com/articles/cm-shri-school-admission-test-result-2025 Thank You.
This ebook serves as a valuable study guide for NEET 2025 exam.
This e-book offers NEET PYQ and serves as an indispensable NEET study material.
As per latest syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters
As per latest syllabus. Chemistry formulas, equations, & laws of class 11 & 12th chapters
As per latest 2024 syllabus. Study 40% syllabus and score upto 100% marks in JEE
As per latest syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters