Find the best tutors and institutes for Class 10 Tuition
Q5(i):
Classify the following as linear, quadratic and cubic polynomials:
(i) $x^2 + x$
Solution :
Given Polynomial & Initial Setup
We are tasked with classifying the following algebraic expression based on its degree:
$P(x) = x^2 + x$
Step 1: Theoretical Foundation of Polynomial Classification
In algebra, polynomials in a single variable are classified according to their degree. [Per the Fundamental Theorem of Algebra and polynomial definitions], the degree of a polynomial is defined as the highest exponent (power) of the variable present in the expression with a non-zero coefficient.
- Linear Polynomial: A polynomial of degree $1$. Standard form: $ax + b$ (where $a \neq 0$).
- Quadratic Polynomial: A polynomial of degree $2$. Standard form: $ax^2 + bx + c$ (where $a \neq 0$).
- Cubic Polynomial: A polynomial of degree $3$. Standard form: $ax^3 + bx^2 + cx + d$ (where $a \neq 0$).
Step 2: Term-by-Term Analysis of $P(x)$
We decompose the given polynomial $P(x) = x^2 + x$ into its constituent terms to identify the exponents of the variable $x$:
- First Term ($x^2$): The variable $x$ is raised to the power of $2$.
- Second Term ($x$): The variable $x$ is implicitly raised to the power of $1$ (since $x = x^1$).
Comparing the exponents $\{2, 1\}$, the maximum value is $2$. Therefore, the highest power of the variable $x$ in the polynomial is $2$.
Step 3: Classification and Geometric Justification
Since the highest exponent is $2$, the degree of the polynomial $P(x) = x^2 + x$ is exactly $2$. By definition, a polynomial of degree $2$ is classified as a quadratic polynomial.
[Geometrically, a quadratic polynomial represents a parabola when graphed on a Cartesian plane. The presence of the $x^2$ term dictates this parabolic curvature, distinguishing it from the straight line of a linear polynomial or the inflection curve of a cubic polynomial.]
Figure 1: The parabolic graph of $y = x^2 + x$ confirms its quadratic nature.
Final Solution: The highest power of the variable $x$ in the expression $x^2 + x$ is $2$. Therefore, it is classified as a quadratic polynomial.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.1
- Q1(i): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (i) $4x^2 – 3x + 7$
- Q1(ii): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (ii) $y^2 + \sqrt{2}$
- Q1(iii): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iii) $3\sqrt{t} + t\sqrt{2}$
- Q1(iv): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iv) $y + \frac{2}{y}$
- Q1(v): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (v) $x^{10} + y^3 + t^{50}$
- Q2(i): Write the coefficients of $x^2$ in each of the following: (i) $2 + x^2 + x$
- Q2(ii): Write the coefficients of $x^2$ in each of the following: (ii) $2 – x^2 + x^3$
- Q2(iii): Write the coefficients of $x^2$ in each of the following: (iii) $\frac{\pi}{2}x^2 + x$
- Q2(iv): Write the coefficients of $x^2$ in each of the following: (iv) $\sqrt{2}x - 1$
- Q3: Give one example each of a binomial of degree 35, and of a monomial of degree 100.
- Q4(i): Write the degree of each of the following polynomials: (i) $5x^3 + 4x^2 + 7x$
- Q4(ii): Write the degree of each of the following polynomials: (ii) $4 – y^2$
- Q4(iii): Write the degree of each of the following polynomials: (iii) $5t – \sqrt{7}$
- Q4(iv): Write the degree of each of the following polynomials: (iv) $3$
- Q5(ii): Classify the following as linear, quadratic and cubic polynomials: (ii) $x – x^3$
- Q5(iii): Classify the following as linear, quadratic and cubic polynomials: (iii) $y + y^2 + 4$
- Q5(iv): Classify the following as linear, quadratic and cubic polynomials: (iv) $1 + x$
- Q5(v): Classify the following as linear, quadratic and cubic polynomials: (v) $3t$
- Q5(vi): Classify the following as linear, quadratic and cubic polynomials: (vi) $r^2$
- Q5(vii): Classify the following as linear, quadratic and cubic polynomials: (vii) $7x^3$
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Polynomials
Govindpuram, Ghaziabad
20 yrs of
Exp
1 student
34 hrs of classes completed
Online Classes
Student's Home
Tutor's home
I teach mathematics and science for class 10th students. In science I teach physics chemistry and biology. My aim is to teach the students in way so that they score 100percent in boards.
Super Tutor
New Friends Colony, Delhi
10 yrs of
Exp
41 students
4626 hrs of classes completed
500
per hour
Online Classes
Hi, I'm Ravi Shankar, a Mathematics educator with 10+ years of teaching experience. I help Class 10 students strengthen their concepts, improve application skills, and prepare systematically for important examinations. At this stage, simply knowing formulas isn't enough. Students need to recognise the appropriate method, interpret questions correctly, solve efficiently and avoid losing marks through preventable mistakes. My classes focus on: • Concept mastery • Algebra and functions • Geometry • Trigonometry • Statistics and probability • Application-based problems • Multi-step questions • Past-paper practice • Error analysis • Time management and exam strategy I use regular assessments to identify recurring mistakes and areas requiring additional attention. My approach: Concept → Application → Problem Solving → Exam Practice → Analysis The goal is to help students approach examinations with greater confidence, accuracy, and consistency. Book an introductory session to understand where your child currently stands and what needs to improve.
I highly appreciate Sir’s dedication and clarity in teaching. His systematic approach helped me score confidently in board examinations.
Super Tutor
Choolissery, Thrissur
20 yrs of
Exp
12 students
446 hrs of classes completed
600
per hour
Online Classes
I was a teacher at a CBSE-affiliated school. I have six years of experience teaching on online platforms. I have been conducting Mathematics, Physics, and Chemistry classes for high school and higher secondary students for more than 20 years. I believe I can encourage and motivate students to achieve excellent academic results while inspiring them to develop good values and habits.
Sini ma'am is an excellent maths teacher. I joined her classes at exactly the right time when I needed clarity, and she explained every concept in a very simple and structured way. Her teaching style made even difficult topics easy to understand, and my confidence in maths improved a lot in a short period. She is very patient, supportive, and genuinely cares about her students’ understanding. I’m really grateful for the guidance and clarity she gave me.
Carterpuri, Palam Vihar, Gurgaon
20 yrs of
Exp
500
per hour
Student's Home
I am a professional teacher with more than 25 years of teaching experience in prestigious schools of India.My teaching assures value for money and produce best results.I emphasize on concepts clarity and create natural interest in the subject.
Super Tutor
Sector 1 Greater Noida, Noida
8 yrs of
Exp
6 students
2328 hrs of classes completed
450
per hour
Online Classes
It was a great experience while taking the class tenth private tutions.
Fabulous experience learning with Indrajeet Sir, quickly made me grasp difficult concepts in preparation for my board examinations.
Find more Tutor for Polynomials in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
