Find the best tutors and institutes for Class 10 Tuition
Q5(i):
Factorise :
(i) $x^3 – 2x^2 – x + 2$
Solution :
Initial Setup & Given Polynomial
We are tasked with finding the complete linear factorization of the given cubic polynomial. Let the polynomial be defined as:
$P(x) = x^3 - 2x^2 - x + 2$
Because the degree of the polynomial is $3$ (a cubic polynomial), the Fundamental Theorem of Algebra dictates that it will have exactly $3$ roots in the complex plane, which corresponds to up to $3$ linear factors of the form $(x - r_i)$, where $r_i$ are the roots.
Step 1: Method of Grouping Terms
For polynomials with four terms, the most efficient initial strategy is factorisation by grouping. We partition the polynomial into two distinct binomial groups to identify common algebraic structures.
Group the first two terms and the last two terms:
$P(x) = (x^3 - 2x^2) - (x - 2)$
[Note: A negative sign is factored out from the third and fourth terms, which reverses the sign of the constant term inside the parenthesis, ensuring mathematical equivalence to the original expression.]
Step 2: Extracting the Greatest Common Factor (GCF)
Next, we extract the Greatest Common Factor from each individual binomial group.
- From the first group $(x^3 - 2x^2)$, the GCF is $x^2$. Factoring this out yields: $x^2(x - 2)$.
- From the second group $(x - 2)$, the GCF is simply $1$. Factoring this out yields: $1(x - 2)$.
Substituting these back into the polynomial expression:
$P(x) = x^2(x - 2) - 1(x - 2)$
Step 3: Factoring Out the Common Binomial
Observe that the binomial $(x - 2)$ is now a common factor to both terms in the expression. [Per the Distributive Property of Multiplication over Addition, $ab - cb = (a - c)b$]. We factor out $(x - 2)$:
$P(x) = (x - 2)(x^2 - 1)$
Step 4: Applying Algebraic Identities
The expression is now a product of a linear binomial and a quadratic binomial. The quadratic factor $(x^2 - 1)$ is a classic "Difference of Two Squares".
[Per the standard algebraic identity: $a^2 - b^2 = (a - b)(a + b)$]
By setting $a = x$ and $b = 1$, we can expand the quadratic term:
$x^2 - 1 = (x - 1)(x + 1)$
Substituting this complete factorization back into our equation for $P(x)$:
$P(x) = (x - 2)(x - 1)(x + 1)$
Alternative Verification: The Factor Theorem
To ensure absolute rigor, we can verify this result using the Rational Root Theorem and the Factor Theorem.
The Rational Root Theorem states that any rational root $\frac{p}{q}$ must have $p$ as a factor of the constant term ($2$) and $q$ as a factor of the leading coefficient ($1$). The possible rational roots are $\pm 1, \pm 2$.
Testing $x = 1$:
$P(1) = (1)^3 - 2(1)^2 - (1) + 2 = 1 - 2 - 1 + 2 = 0$
[Per the Factor Theorem, since $P(1) = 0$, $(x - 1)$ is a guaranteed factor of $P(x)$].
Performing polynomial long division or synthetic division of $(x^3 - 2x^2 - x + 2)$ by $(x - 1)$ yields the quotient $(x^2 - x - 2)$. Factoring this resulting quadratic equation yields $(x - 2)(x + 1)$, perfectly corroborating our grouping method.
Visual Representation: Factorization Tree
The following diagram illustrates the hierarchical breakdown of the polynomial into its constituent linear factors.
Final Solution: The complete linear factorization of the polynomial $x^3 - 2x^2 - x + 2$ is $(x - 2)(x - 1)(x + 1)$.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.3
- Q1(i): Determine which of the following polynomials has $(x + 1)$ a factor : (i) $x^3 + x^2 + x + 1$
- Q1(ii): Determine which of the following polynomials has $(x + 1)$ a factor : (ii) $x^4 + x^3 + x^2 + x + 1$
- Q1(iii): Determine which of the following polynomials has $(x + 1)$ a factor : (iii) $x^4 + 3x^3 + 3x^2 + x + 1$
- Q1(iv): Determine which of the following polynomials has $(x + 1)$ a factor : (iv) $x^3 – x^2 – (2 + \sqrt{2})x + \sqrt{2}$
- Q2(i): Use the Factor Theorem to determine whether $g(x)$ is a factor of $p(x)$ in each of the following cases: (i) $p(x) = 2x^3 + x^2 – 2x – 1, g(x) = x + 1$
- Q2(ii): Use the Factor Theorem to determine whether $g(x)$ is a factor of $p(x)$ in each of the following cases: (ii) $p(x) = x^3 + 3x^2 + 3x + 1, g(x) = x + 2$
- Q2(iii): Use the Factor Theorem to determine whether $g(x)$ is a factor of $p(x)$ in each of the following cases: (iii) $p(x) = x^3 – 4x^2 + x + 6, g(x) = x – 3$
- Q3(i): Find the value of $k$, if $x – 1$ is a factor of $p(x)$ in each of the following cases: (i) $p(x) = x^2 + x + k$
- Q3(ii): Find the value of $k$, if $x – 1$ is a factor of $p(x)$ in each of the following cases: (ii) $p(x) = 2x^2 + kx + \sqrt{2}$
- Q3(iii): Find the value of $k$, if $x – 1$ is a factor of $p(x)$ in each of the following cases: (iii) $p(x) = kx^2 – \sqrt{2}x + 1$
- Q3(iv): Find the value of $k$, if $x – 1$ is a factor of $p(x)$ in each of the following cases: (iv) $p(x) = kx^2 – 3x + k$
- Q4(i): Factorise : (i) $12x^2 – 7x + 1$
- Q4(ii): Factorise : (ii) $2x^2 + 7x + 3$
- Q4(iii): Factorise : (iii) $6x^2 + 5x – 6$
- Q4(iv): Factorise : (iv) $3x^2 – x – 4$
- Q5(ii): Factorise : (ii) $x^3 – 3x^2 – 9x – 5$
- Q5(iii): Factorise : (iii) $x^3 + 13x^2 + 32x + 20$
- Q5(iv): Factorise : (iv) $2y^3 + y^2 – 2y – 1$
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Polynomials
Top Tutor
Mathura Road, Delhi
20 yrs of
Exp
14 students
1590 hrs of classes completed
1000
per hour
Online Classes
Tutor's home
For me, teaching is part of learning. As I am teaching I am also learning from my learners as they are diverse. Each learner in the classroom brings his or her unique background and ability. These are the elements that I invest in my teaching. Moreover, I allow learners to be themselves. This is a way of creating a conducive environment for learning.
Its been a really nice teaching experience with Mr.Amit so far. He is very knowledgeable and a patient teacher. He always clarifies the concepts clearly and ensures good understanding.
Top Tutor
Safilguda Lake, Hyderabad
20 yrs of
Exp
6 students
345 hrs of classes completed
500
per hour
Online Classes
Student's Home
Tutor's home
I was using the urban pro for home tuitions queries. It was very useful and productive platform for the job seekers like me. It helps the needed in giving the contacts and mutually. I would like to avail further services.
Super Tutor
Itagalpura Main Rd, Madappanahalli
17 yrs of
Exp
133 students
3602 hrs of classes completed
600
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.
Super Tutor
Sector 5, Gurgaon
8 yrs of
Exp
28 students
2139 hrs of classes completed
450
per hour
Online Classes
Tutor's home
Mathematics is not just a subject—it is the backbone of science and the foundation for logical thinking and problem-solving. With over eleven years of teaching experience and a B.Tech degree, I am dedicated to nurturing students' mathematical abilities while instilling confidence and a love for learning. Recognitions & Achievements ✅ UrbanPro Excellency Awards – Top 10 in Class 10 Tuition (2024) & Top 100 Tutors for 2024 ✅ Teacher Innovation Award by ZIIEI (Sri Aurobindo Society) ✅ Certified by Leading Ed-Tech Platforms – UrbanPro Certification, GuruQ Certification These accolades reflect my commitment to providing high-quality education and delivering exceptional learning outcomes for my students. Why Choose Me as Your Child’s Tutor? As an educator, my primary goal is to empower students with a strong conceptual foundation while ensuring they enjoy the learning process. I believe that every child has unique strengths and learning needs, and my approach is tailored to bring out the best in them. 📌 Comprehensive Academic Support: I provide in-depth coaching in Mathematics, Science, and English for Grades V–X and specialize in Mathematics for Grades XI–XII and Undergraduate Courses. 📌 Expert in Competitive & Entrance Exam Preparation: I have successfully guided students in securing admissions to prestigious institutions and excelling in competitive exams, including: 🎯 School Entrance Exams – Sainik School, Military School, Navodaya Vidyalaya, Gurukul Entrance, Central Hindu School, Rai Sports School 🎯 Competitive & Government Job Exams – Olympiads, RIMC, NTSE, SSC MTS, CHSL, DSSSB 📌 Proven Track Record of Student Success: My students have consistently demonstrated remarkable improvement in grades and conceptual understanding. I maintain an individualized learning plan for each student, ensuring they receive personalized guidance and continuous support. Regular assessments, progress reports, and feedback sessions help track improvement and address learning gaps effectively. My Teaching Methodology & Approach I strongly believe that learning should be an engaging and interactive experience rather than rote memorization. My experiential learning approach is designed to: ✔ Encourage critical thinking and problem-solving skills ✔ Foster conceptual clarity through real-life applications ✔ Enhance creativity and independent learning To achieve this, I incorporate modern teaching tools and methodologies, such as: 🖥 Live Interactive Sessions – Engaging discussions to clarify doubts in real time 📊 PowerPoint Presentations & Animations – To visualize complex concepts easily 🖋 Whiteboard Teaching – Step-by-step breakdown of mathematical problems 📚 Comprehensive Study Materials, Worksheets & Assignments – Reinforcing learning through practice Online & Offline Classes: I conduct online classes via digital platforms and in-person sessions at my institute in Gurgaon. For students requiring additional support, I offer home tuition when feasible. Commitment to Student Success 🌟 Personalized Attention: I ensure that every student gets the individual focus they need to overcome challenges and master concepts. 🌟 Supportive Learning Environment: My classes are structured to be patient, encouraging, and motivating, allowing students to ask questions freely and develop confidence in their abilities. 🌟 Building Lifelong Skills: Beyond academics, I emphasize the development of problem-solving abilities, logical reasoning, and exam strategies that benefit students in their future careers. 🌟 Parental Involvement & Regular Feedback: I believe in open communication with parents to discuss their child’s progress, strengths, and areas for improvement. Regular performance reviews ensure that parents stay informed about their child's academic growth. Invest in Your Child’s Future Today! With a proven track record, structured teaching approach, and a passion for education, I am committed to helping students achieve excellence in Mathematics and beyond. If you are looking for a dedicated, award-winning tutor who can provide high-quality education, personalized guidance, and measurable results, I would be delighted to support your child’s learning journey. 📩 Contact me today to discuss how I can help your child excel!
Good techniques and enjoyed the class. Loved the way he explained things and gave tricks to save time as well.
Top Tutor
Chitragupta Nagar, Patna Sadar
5 yrs of
Exp
9 students
4996 hrs of classes completed
275
per hour
Online Classes
Hello all, I am Utkarsh kumar. As an experienced, qualified teacher and tutor with over 8 years of experience in teaching Maths, Physics, Chemistry and Biology across different boards including CBSE, and State Board. Passionate about solving mathematical problems and chemical equations. Over the years I have helped thousands of students overcome their fear of Maths, Physics Chemistry and Biology. If u wish to learn in interactive and fun manner, feel free to make a contact.
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
