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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.
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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.
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.
Polynomials Class 10 Exercise: 2.1 Total Questions: 1 Page number: 18 |
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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.
Here are the exercise-wise links for the NCERT class 10 chapter 2 Polynomials:
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.
Students can use the following link to check the solutions of science-related questions in the NCERT book in depth.
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)
A polynomial p(x) is an algebraic expression that can be written in the form of
Here
Relationship between zeros and coefficients of a quadratic polynomial
For the quadratic polynomial
Sum of zeros,
Product of zeros
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.
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.
The highest power of the variable that occurs in the polynomial is called the degree of a polynomial.
On Question asked by student community
Hello,
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.
Hello Aspirant,
Here's how you can find it:
School ID Card: Your registration number is often printed on your school ID card.
Admit Card (Hall Ticket): If you've received your board exam admit card, the registration number will be prominently displayed on it. This is the most reliable place to find it for board exams.
School Records/Office: The easiest and most reliable way is to contact your school office or your class teacher. They have access to all your official records and can provide you with your registration number.
Previous Mark Sheets/Certificates: If you have any previous official documents from your school or board (like a Class 9 report card that might have a student ID or registration number that carries over), you can check those.
Your school is the best place to get this information.
Hello,
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).
For notes , refer to NCERT textbooks as they are the primary source for CBSE exams. Many educational websites also provide chapter-wise revision notes and study material that align with the NCERT syllabus. Focus on practicing previous papers and understanding concepts thoroughly.
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