Find the best tutors and institutes for Class 10 Tuition
Q3(ii):
Factorise the following using appropriate identities:
(ii) $4y^2 – 4y + 1$
Solution :
Step 1: Initial Expression Analysis
We are given the quadratic trinomial:
$P(y) = 4y^2 - 4y + 1$
Our objective is to factorise this expression by identifying it as a perfect square. A quadratic trinomial takes the general form $ax^2 + bx + c$. We must analyze the first and last terms to determine if they are perfect squares.
Step 2: Identifying the Appropriate Algebraic Identity
We observe that the signs of the terms alternate ($+, -, +$). This suggests the use of the standard binomial square identity for subtraction [Per the fundamental axioms of polynomial expansion]:
$(a - b)^2 = a^2 - 2ab + b^2$
Step 3: Term-by-Term Transformation & Verification
We must map the terms of our given polynomial to the components of the identity $a^2 - 2ab + b^2$.
- First Term ($a^2$): The term $4y^2$ can be rewritten as the square of a single monomial.
$4y^2 = (2y)^2 \implies a = 2y$ - Third Term ($b^2$): The constant $1$ is a perfect square.
$1 = (1)^2 \implies b = 1$ - Middle Term ($-2ab$): We must verify that the middle term of our polynomial matches $-2ab$ using our derived values for $a$ and $b$.
$-2ab = -2(2y)(1) = -4y$
Since the middle term perfectly matches the given expression ($-4y$), the polynomial is confirmed to be a perfect square trinomial.
Step 4: Applying the Identity
By substituting $a = 2y$ and $b = 1$ back into the structural framework of the identity, we can rewrite the expanded polynomial as a squared binomial:
$4y^2 - 4y + 1 = (2y)^2 - 2(2y)(1) + (1)^2$
$4y^2 - 4y + 1 = (2y - 1)^2$
Step 5: Geometric Verification (Area Model)
To rigorously prove this factorization, we can use an area model. Consider a large square with a total side length of $2y$. Its total area is $(2y)^2 = 4y^2$. If we partition this square into segments of length $(2y - 1)$ and $1$, we can geometrically derive the identity.
From the geometric model, the total area of the square is the sum of its four internal regions:
$Area_{Total} = (2y - 1)^2 + (2y - 1) + (2y - 1) + 1$
$4y^2 = (2y - 1)^2 + 4y - 2 + 1$
$4y^2 = (2y - 1)^2 + 4y - 1$
Isolating the primary region $(2y - 1)^2$ yields:
$(2y - 1)^2 = 4y^2 - 4y + 1$
[This confirms our algebraic factorization is geometrically absolute].
Step 6: Final Factorisation
To express the polynomial fully in its factorised form, we write the squared binomial as the product of two identical linear binomials.
$(2y - 1)^2 = (2y - 1)(2y - 1)$
Final Solution: The factorised form of the polynomial $4y^2 - 4y + 1$ is $(2y - 1)(2y - 1)$.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.4
- Q1(i): Use suitable identities to find the following products: (i) $(x + 4) (x + 10)$
- Q1(ii): Use suitable identities to find the following products: (ii) $(x + 8) (x – 10)$
- Q1(iii): Use suitable identities to find the following products: (iii) $(3x + 4) (3x – 5)$
- Q1(iv): Use suitable identities to find the following products: (iv) $(y^2 + \frac{3}{2}) (y^2 – \frac{3}{2})$
- Q1(v): Use suitable identities to find the following products: (v) $(3 – 2x) (3 + 2x)$
- Q10(i): Factorise each of the following: (i) $27y^3 + 125z^3$ [Hint : See Question 9.]
- Q10(ii): Factorise each of the following: (ii) $64m^3 – 343n^3$ [Hint : See Question 9.]
- Q11: Factorise : $27x^3 + y^3 + z^3 – 9xyz$
- Q12: Verify that $x^3 + y^3 + z^3 – 3xyz = \frac{1}{2}(x + y + z)[(x – y)^2 + (y – z)^2 + (z – x)^2]$
- Q13: If $x + y + z = 0$, show that $x^3 + y^3 + z^3 = 3xyz$.
- Q14(i): Without actually calculating the cubes, find the value of each of the following: (i) $(–12)^3 + (7)^3 + (5)^3$
- Q14(ii): Without actually calculating the cubes, find the value of each of the following: (ii) $(28)^3 + (–15)^3 + (–13)^3$
- Q15(i): Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (i) Area : $25a^2 – 35a + 12$
- Q15(ii): Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (ii) Area : $35y^2 + 13y –12$
- Q16(i): What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (i) Volume : $3x^2 – 12x$
- Q16(ii): What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (ii) Volume : $12ky^2 + 8ky – 20k$
- Q2(i): Evaluate the following products without multiplying directly: (i) $103 \times 107$
- Q2(ii): Evaluate the following products without multiplying directly: (ii) $95 \times 96$
- Q2(iii): Evaluate the following products without multiplying directly: (iii) $104 \times 96$
- Q3(i): Factorise the following using appropriate identities: (i) $9x^2 + 6xy + y^2$
- Q3(iii): Factorise the following using appropriate identities: (iii) $x^2 – \frac{y^2}{100}$
- Q4(i): Expand each of the following, using suitable identities: (i) $(x + 2y + 4z)^2$
- Q4(ii): Expand each of the following, using suitable identities: (ii) $(2x – y + z)^2$
- Q4(iii): Expand each of the following, using suitable identities: (iii) $(–2x + 3y + 2z)^2$
- Q4(iv): Expand each of the following, using suitable identities: (iv) $(3a – 7b – c)^2$
- Q4(v): Expand each of the following, using suitable identities: (v) $(–2x + 5y – 3z)^2$
- Q4(vi): Expand each of the following, using suitable identities: (vi) $(\frac{1}{4}a - \frac{1}{2}b + 1)^2$
- Q5(i): Factorise: (i) $4x^2 + 9y^2 + 16z^2 + 12xy – 24yz – 16xz$
- Q5(ii): Factorise: (ii) $2x^2 + y^2 + 8z^2 – 2\sqrt{2}xy + 4\sqrt{2}yz – 8xz$
- Q6(i): Write the following cubes in expanded form: (i) $(2x + 1)^3$
- Q6(ii): Write the following cubes in expanded form: (ii) $(2a – 3b)^3$
- Q6(iii): Write the following cubes in expanded form: (iii) $(\frac{3}{2}x + 1)^3$
- Q6(iv): Write the following cubes in expanded form: (iv) $(x - \frac{2}{3}y)^3$
- Q7(i): Evaluate the following using suitable identities: (i) $(99)^3$
- Q7(ii): Evaluate the following using suitable identities: (ii) $(102)^3$
- Q7(iii): Evaluate the following using suitable identities: (iii) $(998)^3$
- Q8(i): Factorise each of the following: (i) $8a^3 + b^3 + 12a^2b + 6ab^2$
- Q8(ii): Factorise each of the following: (ii) $8a^3 – b^3 – 12a^2b + 6ab^2$
- Q8(iii): Factorise each of the following: (iii) $27 – 125a^3 – 135a + 225a^2$
- Q8(iv): Factorise each of the following: (iv) $64a^3 – 27b^3 – 144a^2b + 108ab^2$
- Q8(v): Factorise each of the following: (v) $27p^3 – \frac{1}{216} – \frac{9}{2}p^2 + \frac{1}{4}p$
- Q9(i): Verify : (i) $x^3 + y^3 = (x + y) (x^2 – xy + y^2)$
- Q9(ii): Verify : (ii) $x^3 – y^3 = (x – y) (x^2 + xy + y^2)$
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Polynomials
Super Tutor
Sector 5, Gurgaon
14 yrs of
Exp
29 students
2261 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
Kurichimuttom, Kozhencherry
7 yrs of
Exp
4 students
379 hrs of classes completed
350
per hour
Online Classes
I have taught almost 50 students.
The best teacher my daughter have ever got. She can understand everything that she took and also did many question papers. She could write everything that her teacher teaches her and the teacher is very lovely.
Top Tutor
Bajaj Nagar, Nagpur
3 yrs of
Exp
1 student
Online Classes
Student's Home
Coaching is about improvising a student from where he/she is at. Not about converting an already bright student into an achiever. It should be obvious. The whole idea of ONE ON ONE coaching is to improvise and to elimate the problems in a detailed manner, exactly the way a student desires. To connect and respond to a student cannot be everybodys cup of tea. This is where a capable one comes in to fill the gaps where modern day educationalists fail to fill.
Top Tutor
Kaloor, Kochi
12 yrs of
Exp
2 students
2 hrs of classes completed
500
per hour
Online Classes
Student's Home
Tutor's home
With 12 years of teaching experience, I specialize in concept-based instruction, catering to both weaker students and advanced learners. I help struggling students grasp core concepts while providing advanced-level teaching to high achievers, enabling them to consistently reach 95% or higher in their academic performance.
Good class. Sir explained concepts well, i understood the concepts easily, and sir also cleared all my doubts.
Top Tutor
Sector-18, Faridabad
2 yrs of
Exp
5 students
1038 hrs of classes completed
600
per hour
Online Classes
I have approximately 3 years of experience of teaching Mathematics across different boards including CBSE, IGCSE and state board in reputed schools like DPS International. I am post graduate in statistics and honors degree in mathematics from reputed universities like DU. I have helped many students to overcome their fear of Mathematics and Statistics. My aim is to develop a genuine love for the subject in students.
The best math teacher my son has ever had! She explains every concept with clarity, makes learning enjoyable, and ensures he fully understands each topic. Truly inspiring and supportive.
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
