Find the best tutors and institutes for Class 10 Tuition
Q4(vi):
Find the zero of the polynomial in each of the following cases:
(vi) $p(x) = ax, a \neq 0$
Solution :
Initial Setup & Theoretical Foundation
We are given the linear polynomial:
$p(x) = ax$
with the strict mathematical constraint that $a \neq 0$.
Step 1: Formulating the Equation for the Zero
The "zero" (or root) of a polynomial is defined as the specific value of the independent variable $x$ that evaluates the polynomial to zero. [By the Fundamental Theorem of Algebra and the definition of polynomial roots].
To find this value, we must set the polynomial expression equal to zero:
$p(x) = 0$
$ax = 0$
Step 2: Algebraic Isolation of the Variable
To isolate $x$, we must divide both sides of the equation by the coefficient $a$. We are mathematically permitted to perform this operation strictly because the problem explicitly states the condition $a \neq 0$. [Per the Division Property of Equality, division by any non-zero real number is defined and preserves the equality].
$x = \frac{0}{a}$
Since zero divided by any non-zero number is strictly zero [Per the Zero Property of Division]:
$x = 0$
Step 3: Verification of the Root
To ensure absolute analytical accuracy, we substitute our derived root, $x = 0$, back into the original polynomial function:
$p(0) = a(0)$
$p(0) = 0$
Because the polynomial evaluates to $0$, the value $x = 0$ is rigorously verified as the correct zero of the polynomial.
Graphical Representation & Geometric Interpretation
Geometrically, the zero of a polynomial represents the exact x-coordinate where the graph of the function intersects the x-axis (where $p(x) = 0$). For any non-zero real value of $a$, the linear equation $p(x) = ax$ represents a straight line passing directly through the origin $(0,0)$. The steepness and direction of the line depend on $a$, but the x-intercept remains invariant at the origin.
Final Solution: The zero of the polynomial $p(x) = ax$ (where $a \neq 0$) is $x = 0$.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.2
- Q1(i): Find the value of the polynomial $5x – 4x^2 + 3$ at (i) $x = 0$
- Q1(ii): Find the value of the polynomial $5x – 4x^2 + 3$ at (ii) $x = –1$
- Q1(iii): Find the value of the polynomial $5x – 4x^2 + 3$ at (iii) $x = 2$
- Q2(i): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (i) $p(y) = y^2 – y + 1$
- Q2(ii): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (ii) $p(t) = 2 + t + 2t^2 – t^3$
- Q2(iii): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (iii) $p(x) = x^3$
- Q2(iv): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (iv) $p(x) = (x – 1) (x + 1)$
- Q3(i): Verify whether the following are zeroes of the polynomial, indicated against them. (i) $p(x) = 3x + 1, x = –\frac{1}{3}$
- Q3(ii): Verify whether the following are zeroes of the polynomial, indicated against them. (ii) $p(x) = 5x – \pi, x = \frac{4}{5}$
- Q3(iii): Verify whether the following are zeroes of the polynomial, indicated against them. (iii) $p(x) = x^2 – 1, x = 1, –1$
- Q3(iv): Verify whether the following are zeroes of the polynomial, indicated against them. (iv) $p(x) = (x + 1) (x – 2), x = – 1, 2$
- Q3(v): Verify whether the following are zeroes of the polynomial, indicated against them. (v) $p(x) = x^2, x = 0$
- Q3(vi): Verify whether the following are zeroes of the polynomial, indicated against them. (vi) $p(x) = lx + m, x = –\frac{m}{l}$
- Q3(vii): Verify whether the following are zeroes of the polynomial, indicated against them. (vii) $p(x) = 3x^2 – 1, x = –\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}$
- Q3(viii): Verify whether the following are zeroes of the polynomial, indicated against them. (viii) $p(x) = 2x + 1, x = \frac{1}{2}$
- Q4(i): Find the zero of the polynomial in each of the following cases: (i) $p(x) = x + 5$
- Q4(ii): Find the zero of the polynomial in each of the following cases: (ii) $p(x) = x – 5$
- Q4(iii): Find the zero of the polynomial in each of the following cases: (iii) $p(x) = 2x + 5$
- Q4(iv): Find the zero of the polynomial in each of the following cases: (iv) $p(x) = 3x – 2$
- Q4(v): Find the zero of the polynomial in each of the following cases: (v) $p(x) = 3x$
- Q4(vii): Find the zero of the polynomial in each of the following cases: (vii) $p(x) = cx + d, c \neq 0, c, d$ are real numbers.
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Polynomials
Top Tutor
Uttam Nagar, Delhi
9 yrs of
Exp
5 students
312 hrs of classes completed
300
per hour
Online Classes
Tutor's home
teach them with practical and theory base method
Top Tutor
Patthri Bagh, Dehradun
6 yrs of
Exp
1 student
3 hrs of classes completed
500
per hour
Online Classes
I worked as a Computer teacher in Kendriya Vidyalaya. I worked for a software company Synapse Valutech Private Limited as a Java developer. I worked for so many training centers for Java-like DUCAT, High Technologies, Sky Infotech. Special batches for freshers in JAVA and for working professionals. Special Batches for BA, XI-XII for History, Geography, Economics, Indian Polity, Environment, SSC Maths, English, GS. Vast experience in teaching for class 1 to 10( Science, Maths, SST, Hindi, English Grammar, Computer Science or IT). I have students of DPS Vasant Kunj, DAV Vasant Kunj, Mother's International, Bhatnagar International school, DU and JNU.
Top Tutor
Malkajgiri D.D.Colony, Hyderabad
20 yrs of
Exp
2 students
1338 hrs of classes completed
400
per hour
Online Classes
I was teaching in Government and Government aided schools. I Providing supervision & consultation for students, in the form of academic advising, students’ support, recruitment, courses and programs. Also, participating in student & staff social and cultural activities. PPP Approach to make the students understand the topics. Unified Approach to make the students to work independently and collaboratively. Heuristic Approach to make the students to understand theories through lab practical. Use few other contemporary approaches not limited to Brain Storming, Mind Mapping, Concept Mapping, Snowballing etc.,
Super Tutor
Itagalpura Main Rd, Madappanahalli
17 yrs of
Exp
133 students
3628 hrs of classes completed
500
per hour
Online Classes
I gained extensive experience in delivering high-quality education following the CBSE/ICSE curriculum. My key responsibilities included: Classroom Teaching: Effectively taught Mathematics, Physics and Chemistry to students of classesX, ensuring alignment with the CBSE/ ICSE syllabus and fostering a deep understanding of key concepts. Curriculum Planning: Designed and implemented lesson plans that adhered to the CBSE guidelines, ensuring comprehensive coverage of topics while keeping in mind the varying learning needs of students. Student Assessment: Created and administered tests, quizzes, and assignments in accordance with the CBSE exam pattern. Monitored student performance and provided constructive feedback to facilitate continuous improvement. Interactive Learning: Encouraged an interactive classroom environment, integrating various teaching aids such as multimedia presentations, hands-on activities, and group discussions to enhance student engagement and understanding. Preparation for Board Exams: Provided focused support for students appearing for their CBSE/ICSE board exams by conducting regular revision sessions, mock tests, and personalized guidance to help them excel. Parental Communication: Maintained regular communication with parents regarding student progress, attendance, and overall development. Organized parent-teacher meetings to ensure a collaborative approach to student growth. Co-curricular Involvement: Actively participated in school events, projects, and extracurricular activities, fostering holistic development among students. Professional Development: Engaged in continuous professional development through workshops and CBSE training programs to stay updated with the latest educational practices and policies.
I have approached Sai sir for my daughter Maths tuition. The way you are interacting with student making learning so interesting and explaining everything was just outstanding. Your support and encouragement have given me the confidence.Thank you so much sir for your patience. Your teaching skills are very unique so that any student can easily grasp and enjoy learning.
Top Tutor
Sultanpur Viran, Sultanpur Viran
9 yrs of
Exp
9 students
1105 hrs of classes completed
1000
per hour
Online Classes
Student's Home
Tutor's home
I have a teaching experience of 9+ years. Currently teaching in St. Mary's Convent School. My way of teaching is simple and friendly as per the requirement. I lay emphasis on the basics because that only helps to make the concepts more easier to understand.
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
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
