Find the best tutors and institutes for Class 10 Tuition
Q5(v):
Classify the following as linear, quadratic and cubic polynomials:
(v) $3t$
Solution :
Given Polynomial & Initial Setup
We are given the following algebraic expression:
$P(t) = 3t$
Our objective is to classify this polynomial into one of three distinct algebraic categories: linear, quadratic, or cubic. The classification of a polynomial is strictly determined by its degree.
Step 1: Identify the Variable and its Exponents
In the given polynomial $P(t) = 3t$, the single independent variable present is $t$. To determine the nature of the polynomial, we must inspect the exponents attached to this variable.
The term $3t$ can be explicitly rewritten by revealing its hidden exponent:
$P(t) = 3t^1$
[Per the Fundamental Laws of Exponents, any variable $x$ written without a visible power is mathematically understood to have an exponent of $1$, i.e., $x = x^1$].
Step 2: Determine the Degree of the Polynomial
The degree of a polynomial in one variable is defined as the highest power (exponent) of the variable in that expression.
- The only term containing the variable is $3t^1$.
- The exponent of $t$ in this term is $1$.
Therefore, the highest power of the variable $t$ is $1$. This establishes that the degree of the polynomial $P(t) = 3t$ is exactly $1$.
Step 3: Apply Polynomial Classification Criteria
Polynomials are universally classified based on their degree [By the Fundamental Theorem of Algebra and standard algebraic nomenclature]. The classification matrix is as follows:
| Degree | Classification | Standard Form |
|---|---|---|
| $1$ | Linear | $ax + b$ (where $a \neq 0$) |
| $2$ | Quadratic | $ax^2 + bx + c$ (where $a \neq 0$) |
| $3$ | Cubic | $ax^3 + bx^2 + cx + d$ (where $a \neq 0$) |
Because the degree of $3t$ is $1$, it perfectly satisfies the condition for a linear polynomial. It matches the standard linear form $at + b$ where the leading coefficient $a = 3$ and the constant term $b = 0$.
Step 4: Geometric Verification (Visualizing Linearity)
A defining characteristic of a linear polynomial is that its graph forms a perfectly straight line [Per Cartesian Coordinate Geometry]. Below is the precise geometric representation of $P(t) = 3t$, demonstrating a constant rate of change (slope $m = 3$).
Final Solution: The given polynomial $3t$ has a highest exponent (degree) of $1$. Therefore, it is classified as a linear 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(i): Classify the following as linear, quadratic and cubic polynomials: (i) $x^2 + x$
- 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(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
Top Tutor
Mira Road, Mira-Bhayandar
10 yrs of
Exp
7 students
84 hrs of classes completed
600
per hour
Student's Home
Tutor's home
Math and physics teacher. Expert in cbse education and assessment patten. Been an cbse examiner for 5 continuous years. Aware of the curriculum of ssc, icse and cbse grade 7-10 math and physics. Tutoring since 2006 and been quite successful in giving results.
I would like to say the class was really helpful. I was able to catch a hold on various topics taught in class. Sir also has a very kind nature and he helped us in various way. Overall I loved the class.
Top Tutor
Nerul, Mumbai
10 yrs of
Exp
5 students
640 hrs of classes completed
450
per hour
Online Classes
Student's Home
Tutor's home
I have specifically guided Class 10 students for board exams, with a strong focus on NCERT syllabus, and topics like Real Numbers, Polynomials, Quadratic Equations, and Statistics. I provide regular tests, personalized feedback, and exam-oriented strategies to improve accuracy and speed. Many of my students have shown remarkable improvement and scored above 90% in their board exams.
It was an amazing experience learning and understanding maths with Saurav sir. He really helped me overcome my fear of maths and helped me score good marks in a subject that previously scared me. It really shows how dedicated he is as a teacher. Really good experience!
Top Tutor
Sonari, Jamshedpur
1 yrs of
Exp
4 students
657 hrs of classes completed
300
per hour
Online Classes
Student's Home
Tutor's home
I have a Science background completed my electrical engineering.... I am giving home tuition for 2 years.... I am also preparing for UPSC civil service exam.
Wonderful experience part one month. Look forward to taking more classes with sir. Like the dedicated teaching style.
Top Tutor
Sector-16, Faridabad
1 yrs of
Exp
4 students
532 hrs of classes completed
400
per hour
Online Classes
Student's Home
Tutor's home
I have taught my two sons when they were in class 10. They manged to clear the IIT JEE exam in 1st attempt. I teach by example, which helps the students to visualize the concepts and ideas. I believe in participative learning where the student also gives their viewpoints. This is how they start enjoying the subject and their performance level increases automatically.
My experience so far is great with Sir. He teaches topics very well. He also focuses on the practical applications of the topics.
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.
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
