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
Shankar Vihar, Delhi
18 yrs of
Exp
Student's Home
We are teaching 8th, 9th,10th,11th,12th for the last 12 years and providing 95%+ in our subjects. We are giving best results in DELHI CANTT and South Delhi region.
Top Tutor
Jahangirpura, Surat
15 yrs of
Exp
18 students
1954 hrs of classes completed
250
per hour
Online Classes
I love to teach students the concepts of Maths and Science. I believe in concept clearance rather than rote learning. Teaching is my passion. Many students have achieved 95 % and more in past years and and I believe that if you join and work hard sincerely, you can also achieved such scores.
Good teaching concept and clear explanation of concepts. Keeps students motivated and focused and ensures students understand the basics well. Very supportive and encouraging.
Top Tutor
Vesu, Surat
20 yrs of
Exp
18 students
1271 hrs of classes completed
500
per hour
Online Classes
I have experience teaching Mathematics to Grade 10 students with a focus on strengthening fundamental concepts and preparing them for board examinations. I teach important topics such as Real Numbers, Polynomials, Pair of Linear Equations in Two Variables, Quadratic Equations, Arithmetic Progressions, Trigonometry, Coordinate Geometry, Statistics, and Probability. My teaching approach involves explaining concepts clearly, solving problems step by step, and giving regular practice from NCERT and previous years’ board exam papers. I also conduct tests, revision classes, and doubt-clearing sessions to track students’ progress. My goal is to make mathematics simple and interesting while helping students build confidence and achieve good results in their Grade 10 board exams.
Very good teacher. Patient with the students and understands where they are confused quickly. Motivates the students to be better too.
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.
Top Tutor
Virupakshapura, Bangalore
19 yrs of
Exp
7 students
427 hrs of classes completed
500
per hour
Online Classes
Tutor's home
I've been teaching class 10 CBSE students for over 12 years now and have improved their ability to grasp concepts faster and implement the concepts through real examples. This method has been useful to ignite a liking for the subject and get better grades.
I'm really grateful to have Rajitha ma'am as my maths teacher. She teaches so well, she explains and clarifies doubts clearly that I'm now able to solve tough sums. The tricks you taught me have helped me in my exams. Thank you so much ma'am for helping me score better in my math exams.
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
