Find the best tutors and institutes for Class 10 Tuition
Q8(ii):
Factorise each of the following:
(ii) $8a^3 – b^3 – 12a^2b + 6ab^2$
Solution :
Initial Setup: The Polynomial Expression
We are tasked with factorising the following algebraic expression:
$P(a, b) = 8a^3 - b^3 - 12a^2b + 6ab^2$
Step 1: Identifying the Relevant Algebraic Identity
By observing the degree and the signs of the terms in the polynomial, we note that it consists of four terms: two perfect cubes ($8a^3$ and $-b^3$) and two cross-terms. This structure strongly suggests the expansion of the cube of a binomial difference. [Per the standard algebraic identities for polynomials], the relevant formula is:
$(x - y)^3 = x^3 - y^3 - 3x^2y + 3xy^2$
Step 2: Structural Mapping and Term Transformation
To apply the identity, we must express each term of the given polynomial in the exact form of the identity's expansion. We will determine the base values for $x$ and $y$ by taking the cube roots of the perfect cube terms.
- First term ($x^3$): The term $8a^3$ can be rewritten as a perfect cube. Since $2^3 = 8$, we have $8a^3 = (2a)^3$. Thus, we set $x = 2a$.
- Second term ($-y^3$): The term $-b^3$ can be rewritten as $-(b)^3$. Thus, we set $y = b$.
Now, we must verify if the remaining terms ($-12a^2b$ and $6ab^2$) perfectly match the $-3x^2y$ and $+3xy^2$ components of the identity using our established values for $x$ and $y$.
- Third term ($-3x^2y$): Substituting $x = 2a$ and $y = b$:
$-3(2a)^2(b) = -3(4a^2)(b) = -12a^2b$.
[This perfectly matches the third term of our given polynomial.] - Fourth term ($+3xy^2$): Substituting $x = 2a$ and $y = b$:
$+3(2a)(b)^2 = 6ab^2$.
[This perfectly matches the fourth term of our given polynomial.]
Visualizing the Algebraic Mapping
The following diagram illustrates the structural equivalence between the given polynomial and the standard algebraic identity.
Step 3: Synthesizing the Factored Form
Since all terms of the polynomial $8a^3 - b^3 - 12a^2b + 6ab^2$ map flawlessly to the expansion of $(x - y)^3$ where $x = 2a$ and $y = b$, we can condense the expanded polynomial back into its factored binomial form.
$8a^3 - b^3 - 12a^2b + 6ab^2 = (2a)^3 - (b)^3 - 3(2a)^2(b) + 3(2a)(b)^2$
$= (2a - b)^3$
Step 4: Expanding into Linear Factors
Factorisation requires expressing the polynomial as a product of its irreducible linear factors. The exponent $3$ indicates that the binomial $(2a - b)$ is multiplied by itself three times.
$(2a - b)^3 = (2a - b)(2a - b)(2a - b)$
Final Solution: The completely factorised form of the polynomial $8a^3 - b^3 - 12a^2b + 6ab^2$ is $(2a - b)(2a - b)(2a - b)$.
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(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(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
Top Tutor
New Town, Kolkata
15 yrs of
Exp
40 students
55 hrs of classes completed
200
per hour
Online Classes
I am a strong believer in concept-based gain of knowledge. I start from scratch while teaching the very basic concepts and take my students to a level where they have a deep understanding of the subject. Once they understand the concept, the next step is that practice a lot of questions/numerical/problem in the class itself as practice is the key to retain knowledge. I always refer to prescribed books by CBSE/ICSE board but always encourage students to refer additional 1 or 2 books based on their speed to practice. In last I provide them enough homework to practice at home and send me the doubts before the next class. Also for 10th students, we prepare right from beginning by covering the previous year questions and this approach till date have been fruitful as most of my students excel in the board exams
My son is taking Math and Science Tuition with Rakesh sir for last 2 weeks. I am very much satisfied with his teaching techniques. He uses visual aids which helps the student in understanding the concepts in a better way.
Top Tutor
Electronic City, Bangalore
16 yrs of
Exp
3 students
185 hrs of classes completed
500
per hour
Online Classes
Student's Home
Tutor's home
Mathematics is a subject of understanding in detail by going behind the formula and learning the very basics of it. It's the easiest subject when you understand it. I assure good results with distinctive improvement in maths' understanding.
Top Tutor
Sector - 106, Noida
7 yrs of
Exp
17 students
2094 hrs of classes completed
350
per hour
Online Classes
I am a dedicated mathematics tutor with seven years of experience teaching students from grades 8 to 10 across IB, IGCSE, ICSE, and CBSE boards. I understand that many students struggle with math due to fear, lack of conceptual clarity, or difficulty in applying formulas. My approach focuses on simplifying complex concepts through real-life examples, step-by-step problem-solving, and interactive techniques to make learning engaging. I emphasize building a strong foundation and boosting students' confidence, ensuring they not only understand math but also enjoy it as a subject. I have found that students sometimes lack skills and sometimes they are not very well focused , so I can easily point these out
Govindpuram, Ghaziabad
20 yrs of
Exp
1 student
34 hrs of classes completed
Online Classes
Student's Home
Tutor's home
I teach mathematics and science for class 10th students. In science I teach physics chemistry and biology. My aim is to teach the students in way so that they score 100percent in boards.
Top Tutor
Jayanagar, Bangalore
18 yrs of
Exp
28 students
2306 hrs of classes completed
500
per hour
Online Classes
Private teacher. Classes will be taken for science also but only for Physics and Chemistry. I have been teaching for Class 10 for the past 15 years. Focus will be given to each student regarding their difficult concepts. Previous year question papers will be discussed, along with practice worksheets that will be given for each chapter. Coaching will be given to score good marks in respective board exams by giving them tricks to manage the timings
Wonderful teacher with enormous knowledge about the subject with a smile every time while teaching...
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
