Find the best tutors and institutes for Class 10 Tuition
Q10(i):
Factorise each of the following:
(i) $27y^3 + 125z^3$
[Hint : See Question 9.]
Solution :
Initial Setup & Algebraic Analysis
We are tasked with factorising the following binomial expression:
$27y^3 + 125z^3$
To factorise this polynomial, we must first analyze the coefficients and the degrees of the variables to identify any underlying algebraic structures. We observe that both terms are perfect cubes.
Step 1: Expressing Terms as Perfect Cubes
We determine the cube roots of the numerical coefficients and the variables:
- For the first term: The prime factorization of $27$ is $3 \times 3 \times 3 = 3^3$. Therefore, $27y^3$ can be rewritten as $(3y)^3$ [Applying the power of a product rule: $x^n y^n = (xy)^n$].
- For the second term: The prime factorization of $125$ is $5 \times 5 \times 5 = 5^3$. Therefore, $125z^3$ can be rewritten as $(5z)^3$.
Rewriting the original expression, we get:
$(3y)^3 + (5z)^3$
Step 2: Stating the Relevant Algebraic Identity
The expression is now explicitly in the form of the sum of two cubes, $a^3 + b^3$. [Per the fundamental algebraic identity for the sum of cubes, derived from the expansion of $(a+b)^3 - 3ab(a+b)$], we know that:
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
Step 3: Substitution and Expansion
By mapping our terms to the identity, we set $a = 3y$ and $b = 5z$. Substituting these into the right-hand side of the identity yields:
$(3y)^3 + (5z)^3 = (3y + 5z) \left[ (3y)^2 - (3y)(5z) + (5z)^2 \right]$
Step 4: Simplification of the Quadratic Factor
We must now rigorously simplify each term inside the second set of parentheses (the quadratic trinomial factor):
- Square of the first term: $(3y)^2 = 3^2 \cdot y^2 = 9y^2$
- Product of the two terms: $(3y)(5z) = (3 \cdot 5)(y \cdot z) = 15yz$
- Square of the second term: $(5z)^2 = 5^2 \cdot z^2 = 25z^2$
Replacing the unsimplified terms in our expanded equation with these calculated values, we obtain the final factorised expression:
$(3y + 5z)(9y^2 - 15yz + 25z^2)$
Final Solution: The completely factorised form of $27y^3 + 125z^3$ is $(3y + 5z)(9y^2 - 15yz + 25z^2)$.
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(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(ii): Factorise the following using appropriate identities: (ii) $4y^2 – 4y + 1$
- 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
Badarpur, Delhi
11 yrs of
Exp
145 students
3602 hrs of classes completed
200
per hour
Online Classes
Tutor's home
I am utterly grateful to Akshay sir for his unending support and sincere guidance. He was such a big pillar of support in my board exams and his style of teaching is truly exceptional!!!! His notes, his previous year questions, his tips and his advices are really helpful for the board exams. I was able to score 97.2% in my boards because of his style of teaching. In fact all the topics he marked as important came in the board exams! Moreover he is a very hardworking and dedicated teacher with an unending passion to support and tutor children. I really am grateful to you Akshay sir Regards Nirvaan Rai Batch 2025-26
Super Tutor
Kodambakkam, Chennai
15 yrs of
Exp
57 students
3744 hrs of classes completed
800
per hour
Online Classes
With a decade of experience in teaching mathematics, physics, and chemistry for students from grades 8 through 12 across CBSE, IB, and ICSE boards, I offer a comprehensive educational approach that caters to a diverse range of curricula and learning needs. My academic background includes a BE and an MBA, equipping me with both technical and managerial skills that enhance my teaching methodology. Throughout my career, I have specialized in home tutoring, where I have developed a personalized approach to education that focuses on each student's unique needs and strengths. My sessions are designed to be interactive and engaging, fostering a learning environment where students feel supported and motivated. I conduct both online and offline classes, each lasting one hour, and utilize a variety of teaching tools to facilitate learning. I employ a whiteboard to visually explain concepts and provide question papers to help students practice and prepare for their exams. This hands-on approach ensures that students not only understand theoretical concepts but also develop problem-solving skills and confidence in their subjects. My commitment to delivering high-quality education and my extensive experience make me well-suited to guide students through their academic journey, helping them achieve their full potential in mathematics, physics, and chemistry.
Super Tutor
Nerul Sector 20, Thane
10 yrs of
Exp
62 students
2804 hrs of classes completed
1200
per hour
Online Classes
I have done b.Tech in electrical engineering from IIT KANPUR. I taught one year in akash institute patna, 2 years in fiitjee patna. Currently I am teaching in unacademy. I have taught more than 6000 jee and neet aspirants. In which more than 500 students are selected in iitand aiims. If some one is interested to improve their physics upto jee advanced level take help with me .And see the magic. I can help students in their all type of assignment from any institute like allen, fiitjee, narayna, akash, resonance. I can help students to solve both volume of h.C verma within two month if some one is intrested in solving I.E.Irodov join me and see the next level of physics. My way of teaching is so easy and so advanced that student will realyy enjoy it. Don't waste your time in making decision just join and experience my level
He is a very dedicated teacher with excellent knowledge on topic. He knows Subject very well and teach accordingly to kids.
Super Tutor
Basaveshwara Nagar, Bangalore
12 yrs of
Exp
12 students
3331 hrs of classes completed
600
per hour
Online Classes
With over 15 years of dedicated teaching experience, I am an accomplished and qualified educator specializing in Spoken English, Math, Science, Social Science, and Kannada Language. My expertise extends across various educational boards, including CBSE, ICSE, and state boards. Passionate about unraveling the mysteries of mathematical problems and chemical equations. I have garnered recognition with the prestigious Best Teacher Award for orchestrating state-level Science exams. Having contributed my skills to the esteemed MaxMuller Public School in Bangalore, I am committed to fostering a nurturing and inspiring learning environment. Join me on this educational journey where knowledge meets enthusiasm!
Enrolled my 10-year-old for math tutoring, and the improvement is remarkable. The tutor identifies weak spots quickly, like multiplication tables. Fun drills with timers make practice competitive and enjoyable. She explains errors without blame, turning mistakes into lessons. Custom worksheets align with his CBSE curriculum spot-on. Confidence boost evident; he volunteers answers in class now. Test scores rose steadily over two months. Responsive to parent queries, even on weekends. Affordable rates for such quality one-on-one attention. Planning long-term enrollment—excellent choice!
Super Tutor
Katira Maharaja Hata, Arrah
4 yrs of
Exp
24 students
1769 hrs of classes completed
700
per hour
Online Classes
Tutor's home
I've been teaching Class 10th students of schools from India and abroad. I primarily work with them to prepare for Olympiad, JEE foundation and school exams. I've been teaching students of class 8th-10th Science and Mathematics for more than 3 years. So far, I have taught more than 100 students from India and abroad. I teach international students Mathematics, Physics and Chemistry. I teach almost all boards of India such as CBSE, ICSE and state boards. I also teach IGCSE and IB students. I've my offline coaching institute where I am teaching students of CBSE and state boards who require help in Science and Mathematics. Fee displayed on my profile is for 1Ă—1 classes. For group classes, the fee is substantially reduced.
Find more Tutor for Polynomials in your City
- Delhi Mathematics Tutors
- Bangalore Mathematics Tutors
- Hyderabad Mathematics Tutors
- Chennai Mathematics Tutors
- Kolkata Mathematics Tutors
- Mumbai Mathematics Tutors
- Noida Mathematics Tutors
- Pune Mathematics Tutors
- Gurgaon Mathematics Tutors
- Lucknow Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Jaipur Mathematics Tutors
